Explicit computations of Hilbert modular forms on ℚ(\sqrt{5})

by

Lassina Dembélé

Department of Mathematics and Statistics McGill University, Montreal

September 2002

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for PhD. degree

\begin{matrix}c \\ ◯ \end{matrix}Lassina Dembélé, 2002

Contents

Acknowledgements. i Introduction. 1

Chapter 1. Computing Hilbert modular forms. 5 1. Quaternion algebras: a brief review. 5 2. Classical Hilbert modular forms. 13 3. Hilbert modular forms in the adelic language. 18 4. Hecke operators. 24 5. Automorphic representations. 26 6. The Jacquet-Langlands correspondence. 29 7. Applications. 30 8. Motives. 80 9. Listing of Programs. 83

Chapter 2. Some numerical results on the Oda conjecture. 95 1. A brief review of Hodge structures. 97 2. Cohomology of Hilbert modular surfaces. 101 3. The Hodge structure attached a Hilbert newform. 103 4. Hilbert-Blumenthal abelian varieties. 108 5. The abelian variety associated to a Hilbert newform. 111 6. Computations. 116 7. Program Listing 126 8. Conclusion. 132

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Bibliography 135

Acknowledgements.

I am grateful to my supervisor Prof. H. Darmon who has introduced me to the very beautiful subject of Number Theory. His great patience and wonderful personality have always been a source of encouragement for me. Not only is he a great mathematician but he is also an exceptional as a person.

I thank all my friends G. Alpargu and her husband A. Bourget, T. Jackson, M.–H. Nicole, I. Longhi and C. E. M’lan with whom I shared a lot while studying mathematics at McGill. On a more personal level, I would like to thank my parents from whom I have been away for all these years. It has not always been easy but their moral support has always helped inspire me. I will also like to thank my friends, T. A. Baldé, J. Grah, J.–M. K. D. N’dri, R. Ali Kazmi and P. Reddappagari, for their inavaluable support during my stay at McGill and in Montreal. A special thanks to R. Friedman for all the efforts she put in making sense of my distorted english.

I thank the department of mathematics and statistics of McGill university for their collaboration and for their financial support. A special thanks goes to Prof. N. Elkies who kindly helped me with the Pari search of all elliptic curves in this thesis.

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Abstract. We compute the eigenvalues of the first few Hecke operators T_{𝔭} (N(𝔭) ≤ 100) for all Hilbert modular forms on ℚ(\sqrt{5}) of parallel weight 2 and level of norm less than 500. This is done by exploiting the Jacquet–Langlands correspondence which allows us to transfer such computations to forms on the (totally definite) Hamilton quaternion algebra over ℚ(\sqrt{5}). We use these computations to test numerically a conjecture of Oda on periods of Hilbert modular forms.

Résumé. Nous calculons les valeurs propres des opérateurs de Hecke T_{𝔭} (N(𝔭) ≤ 100) pour toutes les formes modulaires de Hilbert primitives de poids (2, 2) et de niveau de norme plus petite que 500 sur ℚ(\sqrt{5}). Pour ce faire, nous exploitons la correspondence de Jacquet–Langlands qui permet de se ramener au calcul de formes automorphes sur l’algèbre de quaternions de Hamilton sur ℚ(\sqrt{5}). À l’aide de deux examples, nous illustrons la consistance de nos calculs avec la conjecture de Oda sur les périodes de formes modulaire de Hilbert.

Introduction.

Thanks to the works of Wiles [57], Wiles and Taylor [53] and Breuil, Conrad, Diamond and Taylor [7], it is now known that the Taniyama– Shimura–Weil conjecture over ℚ is true; i.e., that every elliptic curve E/ℚ is modular. As a by-product, this put an end to the mathematical saga of Fermat’s Last Theorem that Ribet [43] had already proved to be a consequence of the TSW–conjecture. Before the modularity of all rational elliptic curves was establised, the main numerical results supportive of this conjecture were given by the Antwerp tables [5], later extended by Cremona [14].

Even with the substantial progress accomplished by Wiles in [57] and Skinner–Wiles [50], the hope for a proof of the full TSW–conjecture for elliptic curves over general number fields still seems far away. Meanwhile, motivated by Wiles [57] and related works, Darmon [15] has proposed a strategy to tackle the generalized Fermat Last Theorem. This strategy requires a very good understanding of Hilbert modular forms over the real subfields of certain cyclotomic fields and consists of a series of conjectures. So numerical investigations in order to gather more data supportive of Shimura–Tanyama–Weil conjecture over general number fields still has a role to play as it did in the rational case.

The main ingredient in Cremona [14] is the notion of modular symbols that gives a very concrete description of the first Homology group H_{1}(X_{0}(N), ℤ) of the modular curve of level N amenable to computation. There have been some attempts to generalized this notion to

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higher dimension but the theory is still less elaborate, and is often limited to establishing non–vanishing of L–functions by exhibiting specific modular symbols (see [1, 2, 3, 35, 40]). So to get numerical results about the TSW–conjecture over arbitrary number fields, it is preferable to resort to different computational methods. However, let us mention the work of Oda [40] who has closely investigated this conjecture in the spirit of Deligne [19] in the case of Hilbert modular surfaces. This leads to the so–called Oda’s conjecture. To state it, let F be a totally real field of degree g = [F : ℚ] and 𝔠 an integral ideal in F. We consider the Hilbert modular variety X_{0}(𝔠) and let ℍ_{sp}^{g}(X_{0}(𝔠), ℚ) be its middle cuspidal cohomology. This carries a homogeneous Hodge structure on which the Hecke algebra acts. Let f be a newform and K_{f} the number field generated by its Fourier coefficients.

Conjecture (Oda [24, Chap. VI. sec. 1.]): There is a piece H^{g}(M_{f}, ℚ) of ℍ_{sp}^{g}(X_{0}(𝔠), ℚ), an abelian variety A_{f} of dimension [K_{f} : ℚ] defined over F with good reduction outside of 𝔠, a homomorphism K_{f} → End(A_{f}) ⊗ ℚ, and an isomorphism

⨂_{σ∈Gal(F/ℚ)}H^{1}(A_{f}^{σ}, ℂ) → H^{g}(M_{f}, ℂ)

of K_{f}–rational Hodge structures (A_{f}^{σ} is the abelian variety obtained from A_{f} by the action of σ).

The piece H^{g}(M_{f}, ℚ) is felt to be the Hodge realization of the conjectural motive associated to f.

When g = 2, Oda [40] showed that H^{2}(M_{f}, ℚ) is indeed the tensor product of two K_{f}–rational Hodge structures of weight 1 coming from the isogeny classes of abelian varities A_{f}^{1} and A_{f}^{2}. Unfortunately not much is known about the field of definition of those varieties.

The goal of this thesis was two fold: first discuss an alternative way of computing Hilbert modular newforms of weight 2 on certain quadratic fields and second, see if the forms obtained could provide one with some numerical evidence on the Oda conjecture. Our main focus has been the quadratic field F = ℚ(\sqrt{5}) though there seems to be no major obstacle to extending those computational methods to any real quadratic field of class number one. While we have more or less succeeded with the first goal, the second one should rather be seen more as a work in progress, and the few results we could obtain merely illustrate the mystery of the Oda conjecture.

To explain briefly our method, let F = ℚ(\sqrt{5}) and 𝔠 be an integral ideal. Let f be a newform of weight 2 and level 𝔠 (see Chapter I, sec. 3 for the precise definition). Then, the Hamilton quaternion algebra B over F has class number one. We consider its maximal order R consisting of the icosians; we fix an Eichler order R_{𝔠} of level 𝔠 in R. By the Jacquet–Langlands correspondence, f can be identified with a newform on R_{𝔠, 𝔸}^{×}\B_{𝔸}^{×}/B^{×}. However, R_{𝔠, 𝔸}^{×}\B_{𝔸}^{×}/B^{×} is just the complex analytic space of a zero–dimensional variety. Thus it consists of a finite discrete space. The action of Hecke operators on the complex valued functions on this space is the basic ingredient in the computations. We could compute all the eigenvalues a_{𝔭} of the Hecke operators T_{𝔭} for all prime 𝔭 of norm N(𝔭) ≤ 10000 for any level of norm N(𝔠) ≤ 500, but only the first few (with N(𝔭) ≤ 100) have been listed. It is then easy to compute the periods of such a newform when the plurigenus p_{g} of

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the modular surface X_{0}(𝔠) is equal to 1. We have done so for the two smallest levels; namely, when 𝔠 is one of the prime living above 31 or 41 in F. By direct investigations, we obtain the minimal model of the corresponding elliptic curves.

In the first chapter, we recall the main facts about the Jacquet– Langlands correspondence, then explain in the last section how this is used to perform our computations. In the second chapter, we use the fact that Oda [40] had already establised that integrating a newform along some specific cycles, the corresponding periods are just twists of special values of L–functions, to compute the period moduli of the abelian varieties associated to the newforms of level one of the primes above 31 or 41.

CHAPTER 1 Computing Hilbert modular forms.

In this chapter, we explain how to compute all Hilbert modular forms on F = ℚ(\sqrt{5}) of level an integral ideal 𝔠 and parallel weight 2. Let f be a newform of level 𝔠 and weight 2 on GL_{2}. By the Jacquet-Langlands correspondence, it is associated to an eigenform on the unit group in the Hamilton quaternion algebra over ℚ(\sqrt{5}). The space of modular forms one then gets by making this transfer is simple, hence more suitable for computational purposes. Before delving into those computations, let us recall some facts about Hilbert modular forms, Hecke operators and the Jacquet-Langlands correspondence. We start with a brief review of quaternion algebras.

1. Quaternion algebras: a brief review.

The main references for this section are Vignéras [54] and Reiner [42].

Definition 1.1. A quaternion algebra over F is a central simple F – algebra of rank 4 over F.

Let B be a quaternion algebra over F. By the non-commutativity of B, every element α ∈ B generates a quadratic F –subalgebra L = F [α] of B. The characteristic polynomial of α is of the form x^{2} − tr(α)x + n(α), where tr(α) and n(α) denote respectively the reduced trace and norm of α.

Let τ be the non-trivial automorphism of L. The algebra B viewed as a module over L, under left multiplication is free of rank 2; hence if β ∈ B − L, one has B = L ⊕ Lβ. We can always find a couple (L, a) such that β^{2} = a (see Reiner [42, Chap. VII, sec. 29.]). Then, the multiplication in B is completely determined by the rule βα = α^{τ}β.

Example 1.2. 1) F = ℚ and B = M_{2}(F) the 2 × 2 matrices. Here, we may take L = F × F and a = 1

2) F = ℚ and B = ℚ ⊕ ℚi ⊕ ℚj ⊕ ℚk, where i^{2} = −1, j^{2} = −1 and k = ij = −ji, the Hamilton quaternion algebra. Here, we may take L = ℚ + ℚi, β = j and a = −1.

3) In general, by the structure theorem of Wedderburn for simple algebras, a quaternion algebra B is either a division algebra or isomorphic to the matrix algebra M_{2}(F). If char(F) ≠ 2, we can write B in the form B = F ⊕ F i ⊕ F j ⊕ F k, where i^{2} = a, j^{2} = b, a, b ∈ F^{×} and k = ij = −ji. We denote this algebra by B = (\frac{a, b}{F}). In B = (\frac{a, b}{F}), the conjugate of an element q = x + yi + zj + wk is given by q̄ = x − yi − zj − wk, and the reduced trace and norm respectively by tr(q) = 2x and n(q) = x^{2} + ay^{2} + bz^{2} − abw^{2}.

Definition 1.3. An order R in B is a finite O_{F} –subalgebra of B such that B = F · R.

Example 1.4. Let I be a lattice in B (i.e. a finitely generated O_{F} – module which contains an F –basis of B) and put

O_{l}(I) = {x ∈ B| xI ⊂ I},
O_{r}(I) = {x ∈ B| Ix ⊂ I}.

1. QUATERNION ALGEBRAS: A BRIEF REVIEW. 7

One easily sees that these are O_{F} -algebras contained in B. In fact, they are orders in B. Indeed, let e_{1}, …, e_{4} be an F -basis of B contained in I. For any x, y ∈ B, xy = a_{1}e_{1} + ⋯ + a_{4}e_{4}, for some a_{i} ∈ F. So killing denominators of the a_{i}’s, there exists a ∈ O_{F} such that axy ∈ I. Fixing x and letting y run over the set {e_{1}, …, e_{4}}, we see that there exists a ∈ O_{F} such that ax ∈ O_{l}(I). Hence B = F · O_{l}(I). So we only need see that O_{l}(I) is a finite O_{F} -algebra. Choose an element a ∈ O_{F} ∩ I − {0}. One sees that

O_{l}(I) ⊂ a^{−1}I.

Since a^{−1}I is a finitely generated O_{F} –module, so is O_{l}(I). A similar argument shows that O_{r}(I) is an order in B.

Definition 1.5. An ideal in B is a lattice I ⊂ B. We call O_{l}(I) (resp. O_{r}(I)) the left (resp. right) order of I. We say that I is

(i) two-sided if O_{l}(I) = O_{r}(I); (ii) normal if O_{l}(I) and O_{r}(I) are maximal orders in B; (iii) integral if I is contained in both O_{l}(I) and O_{r}(I); (iv) principal if I is of the form I = O_{l}(I)a = aO_{r}(I). We define the inverse of I to be

I^{−1} = {x ∈ B|IxI ⊂ I}.

The norm n(I) of I is the fractional ideal generated by the norms of all elements in I. When R is an order in B, we say that I is a left (resp. right) R–ideal, if R is contained in O_{l}(I) (resp. O_{r}(I)).

8 1. COMPUTING HILBERT MODULAR FORMS.

By Zorn’s lemma, maximal orders always exist. We say R is an Eichler order if it is the intersection of two maximal orders. We will need the following result of Hijikata later.

Proposition 1.6. [54, Chap. II, Lemma 2.4] Let F be a local field, π a uniformizer of F and R an order in M_{2}(F). Then the following are equivalent.

(i) R is an Eichler order. (ii) There exists a unique pair of maximal orders R_{1} and R_{2} such that R = R_{1} ∩ R_{2}. (iii) There is a unique integer n ∈ ℕ such that R is conjugate to the order

R_{n} = (\begin{matrix}O_{F} & O_{F} \\ π^{n}O_{F} & O_{F} \end{matrix} ) .

(iv) R contains a subring conjugate to (\begin{matrix}O_{F} & 0 \\ 0 & O_{F} \end{matrix} ).

Definition 1.7. Let F be a local field and R an Eichler order in M_{2}(F). The integer n in Proposition 1.6 (iii) is called the level of R.

Definition 1.8. Let R be an order. The different 𝔇(R) of R is the inverse of the dual of R with respect to the trace form. The (reduced) discriminant of R is the ideal disc(R) = n(𝔇(R)) in F.

Proposition 1.9. Let I (resp. R) be an ideal (resp. an order) in B. Then, we have the following.

a) The dual I^{∨} of I is a two–sided ideal in B.

b) 𝔇(R) is an integral two–sided ideal in B.

c) If R is a free O_{F} –module, then disc(R)^{2} = (det(A)), where A = (tr(e_{i}e_{j})) for some O_{F} –basis of R. d) If R′ is an order contained in R, then disc(R′) = disc(R) implies that R′ = R.

Proof. Vignéras [54, Chap. IV, Lemme 4.7. and Corollaire 4.8.] □

Definition 1.10. Let R be an order in B and I, J two left (resp. right) R–ideals. We say that I and J are right (resp. left) equivalent if J = Ix (resp. J = xI) for some non-zero x ∈ B.

Proposition 1.11. Let R be an order in B. The number of left equivalence classes is finite and equal to the number of right equivalence classes. All maximal orders have the same class number.

Definition 1.12. Let R be an order in B. The class number of R is the number of left equivalence classes. The class number of B is the class number of any maximal order in B.

Definition 1.13. Let R and R′ be two orders in B. We say that R and R′ have the same type if they are conjugate by an inner automorphism of B. The number of equivalence classes of maximal orders in B under this equivalence relation is called the type number of B.

Now, let F be a global field and fix a quaternion algebra B over F. Let I be a lattice in B. Then, for any prime 𝔭 of F, the completion I_{𝔭} = I ⊗ O_{𝔭} of I at 𝔭 is a (free) lattice in B_{𝔭}; therefore it is an ideal in B_{𝔭}. The completions (I_{𝔭})_{𝔭} determine an Ô_{F} –module in B̂, where Ô_{F} (resp. B̂) is the finite part of 𝔸 (resp. of B_{𝔸}).

Proposition 1.14. [54, Chap. V, Proposition 5.1.] Let B be a quaternion algebra over F and fix a lattice L in B. Then, the map

I ↦ (I_{𝔭})_{𝔭}

is a bijection between the set of all ideals in B and the set of lattices

{(I_{𝔭})_{𝔭}| I_{𝔭} = L_{𝔭} for almost all 𝔭 in F }.

This Proposition and the very simple structure theorem [54, Theorems 1.1. & 1.3.] of quaternion algebras over local field is the thrust of the global-adelic language. For example, sending (x_{𝔭})_{𝔭} to the ideal determined by the local ideals I_{𝔭} = R_{𝔭}x_{𝔭}, one gets that the quotient R_{𝔸}^{×}\B_{𝔸}^{×} parametrizes the set of left R–ideals which are locally principal. Thus, R_{𝔸}^{×}\B_{𝔸}^{×}/B^{×} parametrizes all left equivalence classes of all R–ideals which are locally principal.

Definition 1.15. We say that B ramifies at 𝔭 if B_{𝔭} is a division ring. We denote by Ram(B) the set of places at which B ramifies.

The following theorem gives a classification of quaternion algebras over a global field. A proof can be found in Vignéras [54, Chap. III.].

Theorem 1.16. [54, Chap. III, Theorem 3.1.] Let B be quaternion algebra over a global field F. Then Ram(B) is a finite set of even cardinality. Conversely, for every finite set S of places of even cardinality, there is a unique quaternion algebra B (up to isomorphism) such that Ram(B) = S.

1. QUATERNION ALGEBRAS: A BRIEF REVIEW. 11

The reduced discriminant of B is defined to be the integral ideal disc(B) ⊂ O_{F} determined by all the primes 𝔭 ∈ Ram(B). This is a well-defined ideal as Ram(B) is finite by the above theorem.

Let R be an Eichler order. Then, for any prime 𝔭 ∈ Ram(B), R_{𝔭} is the unique maximal order in B_{𝔭}; while if 𝔭 ∉ Ram(B), by Hijikata [54, Lemma 2. 4], disc(R_{𝔭}) = (π_{𝔭}^{n_{𝔭}}), where n_{𝔭} is the level of R_{𝔭} and π_{𝔭} a uniformizer of O_{𝔭}.

Definition 1.17. Let R be an Eichler order. The level 𝔑(R) of R is the unique integral ideal determined by the local discriminant disc(R_{𝔭}), for 𝔭 ∉ Ram(B).

By the local nature of the discriminant, we see that

disc(R) = disc(B)𝔑(R).

The class number formula for orders in quaternion algebras can be found in several places. We refer to Vignéras [54, Chap. III, IV & V] and references therein. The following result will be enough for our purpose.

Proposition 1.18. [55] Let F be a number field of degree g and B a totally definite quaternion algebra over F (i.e., Ram(B) contains all the archimedian places of F). Let R be a maximal order in B and h its class number. We fix a set of representatives of all left ideal classes of R, say {I_{1}, …, I_{h}}. For each i, let R_{i} be the right order of I_{i}. We let R_{i}^{×} (resp. R_{i}^{1}) be the group of units (resp. elements of norm 1) in R_{i}. We denote by w_{i} the cardinality of R_{i}^{1}. Then,

ζ_{F} (−1) ∏_{𝔭|disc(B)}(1 − N(𝔭)) = 2^{g} ∑_{i=1}^{h} \frac{[R_{i}^{×} : O_{F}^{×}R_{i}^{1}]}{w_{i}}.

Proof. See Vignéras [55]. □

Example 1.19. Let F = ℚ(\sqrt{5}) and B the Hamilton quaternion algebra (i.e. the unique quaternion algebra, which ramifies only at both infinite places). Put

R = ℤ[ω][e_{1}, e_{2}, e_{3}, e_{4}],

with

e_{1} = \frac{1}{2}(1 − ωi + ω̄j),

e_{2} = \frac{1}{2}(−ωi + j + ω̄k),

e_{3} = \frac{1}{2}(ω̄i − ωj + k),

e_{4} = \frac{1}{2}(i + ω̄j − ωk),

and ω = (1 + \sqrt{5})/2. Then disc(R) = 1, thus R is a maximal order: this is the icosian ring (see Conway and Sloane [13, Chap. VIII, sec. 2.1.]). R^{1} is the icosian group and R^{×} is the semi–direct product of R^{1} with O_{F}^{×} = ℤ[ω]^{×}. Keeping the same notations as in the above proposition, we see that by the structure theorem of finite multiplicative subgroups of the Hamilton quaternion algebra ℍ over the real [54, Chap. I, Théorème 3.7.], the only possibilities for w_{i} are 24, 48 and 120. Again from [54, Chap. I, Théorème 3.7. and Exercice 3.1.], the only possibilities for [R_{i}^{×} : O_{F}^{×}R_{i}^{1}] are 1, 2, 4. Since ζ_{ℚ(\sqrt{5})}(−1) = 1/30, we see that we must necessarily have h = 1, [R_{1}^{×} : O_{F}^{×}R_{1}^{1}] = 1 and w_{1} = 120. That is, B has class number 1.

2. Classical Hilbert modular forms.

The main reference for this section is van der Geer [24].

Through out the rest of this chapter, F is a totally real number field of degree g. We let a → a^{τ_{i}}, i = 1, …, g, be all the real embeddings of F. Also, we let O_{F} be the ring of integers of F, δ its different and 𝔸 its adèle ring. We say that an element a ∈ F is totally positive if a^{τ_{i}} > 0, for all i = 1, …, g. We denote by F^{+} (resp. O_{F}^{+}), the set of all totally positive elements in F (resp. O_{F}). Through out, 𝔟 (resp. 𝔠) will be a fractional (resp. integral) ideal of F.

We consider the reductive group G = Res_{F/ℚ}GL_{2} obtained by restriction of scalars à la Weil, so that for any ℚ-algebra A, G_{A} = GL_{2}(A ⊗_{ℚ} F). We let

GL^{+}(O_{F} ⊕ 𝔟) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) : a, d ∈ O_{F} , b ∈ 𝔟^{−1}, c ∈ 𝔟,
ad − bc ∈ O_{F}^{×}^{+}}.

This is the subgroup of all automorphisms of the projective rank 2 O_{F} - module O_{F} ⊕ 𝔟 having totally positive determinant. We put Γ(𝔟, 𝔠) to be the following subgroup of GL^{+}(O_{F} ⊕ 𝔟).

Γ(𝔟, 𝔠) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL^{+}(O_{F} ⊕ 𝔟) : a, d ∈ O_{F} , b ∈ 𝔟^{−1}, c ∈ 𝔠𝔟\} .

GL^{+}(O_{F} ⊕ 𝔟) can be embedded into GL_{2}^{+}(ℝ)^{g} by making use of the g embeddings τ_{i} : F ↪ ℝ. We recall that GL_{2}^{+}(ℝ)^{g} acts on the gfold product H^{g} of the upper half-plane by extension of the well-known action of GL_{2}^{+}(ℝ) on H by Mobius transformations:

GL_{2}^{+}(ℝ) × H → H
(γ, z) ↦ \frac{az + b}{cz + d},

where γ = (\begin{matrix}a & b \\ c & d \end{matrix}). This action of GL^{+}(ℝ)^{g} factors through PGL^{+}(ℝ)^{g}

and the resulting action (up to a semi-direct factor given by 𝔖_{g}, the group of g-permutations), gives all the biholomorphic transformations of H^{g}. For any two ideals 𝔟_{1} and 𝔟_{2}, GL^{+}(O_{F} ⊕ 𝔟_{1}) and GL^{+}(O_{F} ⊕ 𝔟_{2}) are conjugate (by an inner automorphism) in M_{2}(F) if and only if 𝔟_{1} and 𝔟_{2} belong to the same genus (i.e., there exists some totally positive λ ∈ F and an ideal 𝔟 such that 𝔟_{1} = λ𝔟^{2}𝔟_{2}).

The domain H^{g} is contained in P^{1}(ℂ)^{g}, and the g embeddings τ_{i} : F ↪ ℝ induce an embedding

P^{1}(F) ↪ P^{1}(ℝ)^{g} ⊂ P^{1}(ℂ)^{g}.

GL_{2}^{+}(ℝ)^{g} acts on P^{1}(ℝ)^{g} by fractional linear transformations, and so does GL^{+}(O_{F} ⊕ 𝔟) on P^{1}(F) ⊂ P^{1}(ℝ)^{g}.

Let us fix Γ a subgroup of GL^{+}(O_{F} ⊕ 𝔟) of finite index. The Γequivalent classes of P^{1}(F) are called the cusps of Γ. The class of (1 : 0) is called the cusp ∞. The isotropy group of the cusp ∞ in GL^{+}(O_{F} ⊕ 𝔟) is

GL^{+}(O_{F} ⊕ 𝔟)_{∞} = \{(\begin{matrix}ε_{1} & μ \\ 0 & ε_{2} \end{matrix}) : ε_{1}, ε_{2} ∈ O_{F}^{×}, ε_{1}ε_{2} ∈ O_{F}^{×}^{+}, μ ∈ 𝔟^{−1}\} .

It determines the transformation group

PGL^{+}(O_{F} ⊕ 𝔟)_{∞} = \{(\begin{matrix}ε & μ \\ 0 & 1 \end{matrix}) : ε ∈ O_{F}^{×}^{+}, μ ∈ 𝔟^{−1}\} .

Hence the transformation group corresponding to the isotropy group Γ_{∞} = Γ ∩ GL^{+}(O_{F} ⊕ 𝔟)_{∞} of ∞ in Γ is of the form

Γ̂_{∞} = \{(\begin{matrix}ε & μ \\ 0 & 1 \end{matrix}) : ε ∈ V, μ ∈ M\} ,

where V is a subgroup of O_{F}^{×}^{+}, while M is a ℤ-submodule of 𝔟^{−1}. In fact, since Γ has finite index in GL^{+}(O_{F} ⊕ 𝔟), M is a sublattice of 𝔟^{−1}, and the Dirichlet unit theorem implies that V is a subgroup of O_{F}^{×}^{+} of rank g − 1. We call (M, V) the type of the cusp ∞.

Let k = (k_{1}, …, k_{g}) ∈ ℤ^{g} be a vector with integer coefficients, and f : H^{g} → ℂ a meromorphic function. Take γ = (γ_{1}, …, γ_{g}) ∈

GL_{2}^{+}(ℝ)^{g}, with γ_{i} = (\begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix}) and put

(f||_{k}γ)(z) := (∏_{i=1}^{g} det(γ_{i})^{k_{i}/2}(c_{i}z_{i} + d_{i})^{−k_{i}}) f(γz),

for z = (z_{1}, …, z_{g}) ∈ H^{g}. This defines an action of GL_{2}^{+}(ℝ)^{g} on the set of meromorphic functions on H^{g}.

Definition 1.20. A Hilbert modular form of weight k and level Γ is a holomorphic function f : H^{g} → ℂ such that

f||_{k}γ = f, for all γ ∈ Γ.

Let f : H^{g} → ℂ be a Hilbert modular form. Since it is Γ-invariant, we have in particular

f(z + μ) = f(z), for all z ∈ H^{g}, μ ∈ M.

Therefore, it admits a Fourier expansion of the form

f(z) = ∑_{ν∈M^{∨}} a_{ν}e^{2πiTr(νz)},

where Tr(νz) = ∑_{i=1}^{g} ν^{τ_{i}}z_{i}, and M^{∨} is the dual lattice of M with respect to the trace form. When g > 1, every Hilbert modular form is automatically holomorphic at cusps as the next lemma shows.

Lemma 1.21 (Koecher’s principle). Assume that g > 1. Then, f is holomorphic at the cusp ∞ (hence at all cusps) in the following sense:

a_{ν} ≠ 0 ⇒ ν = 0 or ν ≫ 0.

Proof. Let (M, V) be the type of the cusp ∞, and take ε ∈ V.

Then, γ(ε) = (\begin{matrix}ε & 0 \\ 0 & 1 \end{matrix}) ∈ Γ. Hence, f||_{k}γ(ε) = f. Equating the q–

expansion of both members of this equality, it follows that

a_{εν} = N(ε)^{k/2}a_{ν}, for all ν ∈ M^{∨},

where we use the notation N(ε) = ε_{1}^{k_{1}} ⋯ ε_{g}^{k_{g}}. Now, let us assume that there is a non–zero ν_{0} ∈ M^{∨} not totally positive such that a_{ν_{0}} ≠ 0. Without lost of generality, we can assume that ν_{0}^{τ_{1}} < 0. By the Dirichlet units theorem, we can find ε ∈ V such that

ε^{τ_{1}} > 1 and ε^{τ_{i}} < 1, i = 2, …, g.

2. CLASSICAL HILBERT MODULAR FORMS. 17

We now consider the subseries of f(z) = ∑_{ν∈M^{∨}} a_{ν}e^{2πiTr(νz)} index by the set {ν_{0}ε^{m}, m ∈ ℕ}, in which we put z = (i, …, i). Then

a_{ν_{0}ε^{m}}e^{−2πTr(ν_{0}ε^{m})} = N(ε)^{mk/2}a_{ν_{0}}e^{−2πTr(ν_{0}ε^{m})}.

But, as m → ∞, e^{−2πTr(ν_{0}ε^{m})} ∼ e^{−2πν_{0}^{τ_{1}}(ε^{τ_{1}})^{m}}, and the exponential growth ensures that N(ε)^{mk/2}a_{ν_{0}}e^{−2πTr(ν_{0}ε^{m})} → ∞. Therefore the series does not converge, which is a contradiction. So we must have a_{ν_{0}} = 0. □

Definition 1.22. We say that f is a cusp form if the constant term a_{0} in the Fourier expansion is equal to 0 for any f||_{k}γ, γ ∈ GL_{2}^{+}(F) (i.e., if f vanishes at all cusps).

We will denote by M_{k}(Γ) (resp. S_{k}(Γ)) the space of Hilbert modular forms (resp. cusp forms) of weight k and level Γ.

Corollary 1.23. S_{k}(Γ) = M_{k}(Γ) unless k_{1} = ⋯ = k_{g}.

Proof. Let assume that there is f ∈ M_{k}(Γ) not a cusp form. Then at some cusp σ, the q–expansion must give a_{0} ≠ 0. From

a_{0} = N(ε)^{k/2}a_{0}, for all ε ∈ V,

it follows that we must have N(ε)^{k/2} = 1 for all ε ∈ V. But this is possible only if we have k_{1} = ⋯ = k_{g}. □

Proposition 1.24. (i) M_{k}(Γ) = 0 unless k_{i} ≥ 0 for all i = 1, …, g. (ii) M_{0}(Γ) = ℂ and S_{0}(Γ) = 0.

Proof. van der Geer [24, Chap. I. sec. 6.] □ Example 1.25. Eiseinstein series. Let B be an ideal class and 𝔟 ∈ B. Let k ≥ 2 be even. Put

G_{k, B}(z) = N(𝔟)^{k} ∑_{(c,d)∈P^{1}(𝔞𝔟×𝔟)} N(cz + d)^{−k},

where P^{1}(𝔞𝔟 × 𝔟) = {(c, d) ∈ 𝔞𝔟 × 𝔟|(c, d) ≠ (0, 0)}/O_{F}^{×}. This series does not depend on 𝔟 ∈ B, and it can be shown to be modular form of weight k = (k, ⋯, k) with respect to SL(O_{F} ⊕ 𝔞). As in the one– dimensional case, one can find the expansion at ∞ by making use of Poisson summation (see van der Geer [24, Chap. 1, sec. 6]). We call G_{k, B} the Eiseinstein series of weight k and class B with repect to SL(O_{F} ⊕ 𝔞).

The Petersson inner product on S_{k}(Γ) is given by

⟨f, g⟩ = μ(Γ\H^{g})^{−1} ∫_{Γ\H^{g}} f(z)\overline{g(z)}y^{k}dμ(z),

where dμ(z) = ∏_{j=1}^{g} y_{j}^{−2}dx_{j}dy_{j}, with z_{j} = x_{j} + y_{j}i.

3. Hilbert modular forms in the adelic language.

In this section, we fix a quaternion algebra B over F and let G = Res_{F/ℚ}(B^{×}) be the reductive algebraic group obtained by restriction of scalars. For any F -algebra A, G(A) is the group of all A-valued points of G. In particular, G(𝔸) is the adelization of G, and G_{∞} and G_{f} respectively, its infinite and finite parts. The subgroup of all totally positive elements of G_{∞} will be denoted G_{∞}^{+}. We fix a maximal compact subgroup K_{∞} of G_{∞}^{+} and let K_{f} be a compact open subgroup of G_{f}. Put K = K_{∞}×K_{f}. Let 𝔤 be the Lie algebra of G_{∞}. Then we denote by 𝔤_{ℂ} the complexification of 𝔤, by U(𝔤_{ℂ}) the universal enveloping algebra of 𝔤_{ℂ}, and by Z(𝔤_{ℂ}) the center of U(𝔤_{ℂ}).

Let Φ : G(𝔸) → ℂ be a continuous function. We say that Φ is smooth if it is C^{∞} as a function on G_{∞} and locally constant on G_{f}. We say that it is K–finite to the right if the complex space W_{Φ} = Span({Φ_{k}, k ∈ K}), where Φ_{k} : G(𝔸) → ℂ is the function sending g to Φ_{k}(g) = Φ(gk), is finite dimensional. This means that the representation of K we obtain by right translation of Φ is finite dimensional. The elements in U(𝔤_{ℂ}) can be thought of as differential operators or as distributions (see [23]). As such, they act on Φ in a natural way. So we can define Z(𝔤_{ℂ})–finiteness similarly to K–finiteness. Finally, we say that Φ is slowy increasing if there is a continuous representation ρ : G(𝔸) → GL_{n}(ℂ) such that, for any g ∈ G(𝔸),

|Φ(g)| ≤ C∥ρ(g)∥_{2},

where ∥A∥_{2} = Tr(AA^{t})^{1/2} is the Schur norm.

Definition 1.26. An automorphic form on G(𝔸) of level K_{f} is a smooth function Φ : G(𝔸) → ℂ satisfying the following conditions:

(i) Φ(γgz) = Φ(g), for all γ ∈ G(ℚ), g ∈ G(𝔸) and z ∈ Z_{𝔸}; (ii) Φ is right K–finite and Z(𝔤_{ℂ})–finite; (iii) Φ is slowly increasing. If Φ satisfies the further condition that

∫_{N(𝔸)/N(F)} Φ(ng)dn = 0, for all g ∈ G(𝔸),

where N is the unipotent radical of G, we say that Φ is cuspidal (or a cusp form).

20 1. COMPUTING HILBERT MODULAR FORMS.

When G = Res_{F/ℚ}(GL_{2}), we let K_{∞} = SO(2)^{g}. Then, since K_{∞} is abelian, the finite dimensionality of the representation W_{Φ} implies that it is induced by some character χ of (ℝ/ℤ)^{g}. If χ(θ) = exp(2πiTr(kθ) for θ ∈ ℝ^{g} and k ∈ ℤ^{g}, then

Φ(gr(θ)) = χ(θ)Φ(g), for all g ∈ G(𝔸),

r(θ) = ((\begin{matrix}cos(2πθ_{i}) & − sin(2πθ_{i}) \\ sin(2πθ_{i}) & cos(2πθ_{i}) \end{matrix})) .

In this setting, the center of the universal enveloping algebra is generated by the Casimir operators Δ_{i} corresponding to each archimedian place. Therefore the Z(𝔤_{ℂ})–finiteness of Φ is equivalent to Φ being an eigenvector of each Δ_{i}. Recall that for the Lie group GL_{2}^{+}(ℝ), the Casimir operator Δ is given by

−4Δ = H^{2} + 2RL + 2LR,

where

R = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , L = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , Z = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) .

Definition 1.27. A Hilbert modular form of level K_{f} and weight k ∈ ℤ^{g} is an automorphic form Φ : G(𝔸) → ℂ such that

(i) Φ(gk) = Φ(g), for all k ∈ K_{f} (k_{∞} = 1); (ii) Φ(gr(θ)) = χ(θ)Φ(g), for all g ∈ G(𝔸), r(θ) ∈ K_{∞}, where the character χ is defined as above;

(iii) Δ_{i}Φ = −\frac{k_{i}}{2} (\frac{k_{i}}{2} − 1) Φ, for all i = 1, …, g.

We still assume that G = Res_{F/ℚ}(GL_{2}), and let Y (𝔠) and K_{0}(𝔠) be the following semigroups.

K_{0}(𝔠) = ∏_{𝔭}K_{𝔭}, Y (𝔠) = G(𝔸) ∩ (G_{∞}^{+} × ∏_{𝔭}Y_{𝔭}) ,

where the product is taken over all finite primes of F, and

Y_{𝔭} = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(F_{𝔭}) : b, d ∈ O_{𝔭}, a − 1, c ∈ 𝔠_{𝔭}\} ,

K_{𝔭} = {x ∈ Y_{𝔭} : det(x) ∈ O_{𝔭}^{×}}.

We simply call a Hilbert modular form of level K_{0}(𝔠) and weight k a Hilbert modular form of level 𝔠 and weight k. We will denote by M_{k}(𝔠) (resp. S_{k}(𝔠)) the space of Hilbert modular forms (resp. cusp forms) of weight k and level 𝔠. The connection of the spaces M_{k}(𝔠) (resp. S_{k}(𝔠)) with the classical Hilbert modular forms is obtained as follows: let 𝔠_{λ}, λ = 1, …, h^{+}, be representatives of the narrow ideal classes of F. For each λ = 1, …, h^{+}, take x_{λ} ∈ G_{𝔸}, so that t_{λ} = det(x_{λ}) generates the ideal 𝔠_{λ}. Then, by the strong approximation theorem,

G(𝔸) = ∐_{λ=1}^{h^{+}} G(ℚ)x_{λ}(G_{∞}^{+} × K_{0}(𝔠)),

and we see that

Γ_{λ} = Γ(𝔠_{λ}, 𝔠) = x_{λ}(G_{∞}^{+} × K_{0}(𝔠))x_{λ}^{−1} ∩ G(ℚ).

To each h^{+}–tuple (f_{1}, …, f_{h^{+}}) ∈ ⊕_{λ=1}^{h^{+}}S_{k}(Γ_{λ}), we associate the ℂvalued function f on G(𝔸) defined by

f(γx_{λ}w) = (f_{λ}||_{k}w_{∞})(i), γ ∈ G(ℚ) and w ∈ G_{∞}^{+} × K_{0}(𝔠),

where i = (i, …, i). The function f uniquely determines the h^{+}–tuple (f_{1}, …, f_{h^{+}}).

Proposition 1.28. [25, Chap. 3] The map

⨁_{λ=1}^{h^{+}} S_{k}(Γ_{λ}) → S_{k}(𝔠)

(f_{1}, …, f_{h^{+}}) ↦ f

is an isomorphism of complex vector spaces.

In our general setting, we will denote by A^{k}(G, K_{f}) (resp. A_{0}^{k}(G, K_{f})) the space of all automorphic (resp. cusp) forms. They satisfy the following results, which we refer to [30, Chap. 4] and [6] for proofs.

Theorem 1.29. The space A^{k}(G, K_{f}) is finite dimensional and every element in A_{0}^{k}(G, K_{f}) is square-integrable.

Again, let G = Res_{F/ℚ}(GL_{2}) and Φ ∈ A_{0}(G, K_{f}) be a cusp form. We recall that the standard additive character τ_{0} of A_{ℚ} is given by τ_{0} = ∏_{𝔭} τ_{𝔭}, where

τ_{∞}(x_{∞}) = e^{2πix_{∞}},
τ_{𝔭}(x) = e^{−2πiy}, where x = y + z, y ∈ ℤ[1/p], z ∈ ℤ_{p},

at each prime p. The standard additive character is obtained by composing τ_{0} with the trace map 𝔸 = 𝔸_{F} → 𝔸_{ℚ}. Then, any character of 𝔸/F is of the form τ_{ξ} : x ↦ τ(ξx), ξ ∈ F. For almost all g ∈ G(𝔸),

x ↦ Φ ((\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) g)

is a square-integrable function on 𝔸/F. Hence, it admits the Fourier expansion:

Φ ((\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) g) = ∑_{ξ∈F} W_{ξ}(g)τ(ξx),

3. HILBERT MODULAR FORMS IN THE ADELIC LANGUAGE. 23

where

W_{ξ}(g) = ∫_{𝔸/F} τ̄(ξu)Φ ((\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) g) du.

Definition 1.30. We call the function W_{ξ} : g ↦ W_{ξ}(g) the ξ^{th}-Fourier coefficient of Φ.

Take f ∈ S_{k}(𝔠) and let (f_{1}, …, f_{h^{+}}) be the corresponding h^{+}–tuple of classical Hilbert modular forms. Each component f_{λ} has a q-expansion

f_{λ}(z) = ∑ a_{λ}(ξ)e^{2πiTr(ξz)} (ξ = 0 or 0 ≪ ξ ∈ i(t_{λ})),

where i(t) is the ideal corresponding to the idèle t. Since f_{λ}(εz)ε^{k/2} = f_{λ}(z) for ε ∈ O_{F}^{×}^{+}, we see that a_{λ}(ξ)ξ^{−k/2}, for ξ ≠ 0, depends only on the ideal i(ξ). Now, by definition of the t_{λ}, every non-zero integral ideal of F is of the form i(ξt_{λ}^{−1}) for some unique λ and some totally positive ξ ∈ i(t_{λ}). We define c(𝔞, f) by

c(𝔞, f) = \begin{cases}a_{λ}(ξ)t_{λ}^{−k/2} & if 𝔞 = i(ξt_{λ}^{−1}) is integral \\ 0 & if not. \end{cases}

By mimicking the proof of Lemma 3. 6 in [25, Gelbart], we get that the q-expansion of the cusp form f is given by

W_{ξ} ((\begin{matrix}y & 0 \\ 0 & 1 \end{matrix})) = \{ \begin{matrix}c(𝔞, f)e^{−2πTr(ξy_{∞})} & if 𝔞 = i(ξ)i(t_{λ})^{−1} is integral \\ 0 & otherwise, \end{matrix}

ξ ≫ 0, y_{∞} ≫ 0.

The Dirichlet L–series associated to f is

L(f, s) = ∑_{𝔞} \frac{c(𝔞, f)}{N(𝔞)^{s}}.

It converges for Re(s) large enough.

The Petersson inner product on S_{k}(𝔠) is then given by

⟨f, g⟩ = ∑_{λ=1}^{h^{+}}⟨f_{λ}, g_{λ}⟩.

4. Hecke operators.

We refer the reader to Shimura [48, Chap. 3] and [47, sec. 2], also to Miyake [38, Chap. 2, sec. 2], which contain a very nice description of the local Hecke algebras and their relation to the global one. Though the discussion in Miyake [38] is over ℚ_{p}, it still applies to any p-adic field. In our discussion, we assume that all modular forms have parallel weight, i.e. k = (k, …, k). (See [48] for more details on the nonparallel case). The Hecke algebra, defined by the group K_{0} = K_{0}(𝔠) and the semi-group Y = Y (𝔠) consists of formal finite sums ∑_{y∈Y} c_{y}K_{0}yK_{0}, with y ∈ Y and c_{y} ∈ ℂ. The multiplication law is defined as in [48, Chap. 3, sec. 1]. It acts on M_{k}(𝔠) as follows: for y ∈ Y with K_{0}yK_{0} = ⋃_{j} K_{0}y_{j}, where (y_{j})_{∞} = 1,

(f|K_{0}yK_{0})(x) = ∑_{j} f(xy_{j}), x ∈ G(𝔸).

This is independent of the choice of y′_{j}s. For each integral ideal 𝔞 of F, we define the (normalized) Hecke operator T_{𝔠}(𝔞) on f ∈ M_{k}(𝔠) by

T_{𝔠}(𝔞)f = N(𝔞)^{k/2−1} ∑_{det(i(y))=𝔞}(f|K_{0}yK_{0}),

where the sum is taken over all distinct double cosets K_{0}yK_{0} for which det(i(y)) = 𝔞. Also, for eah integral ideal 𝔞 of K prime to 𝔠, we define the operator S_{𝔠}(𝔞)

S_{𝔠}(𝔞)f = f|K_{0}aK_{0},

4. HECKE OPERATORS. 25

where a is any element of 𝔸^{×} such that 𝔞 = i(a). For convenience, we set S_{𝔠}(𝔞) to be zero when 𝔞 is not prime to 𝔠. As for modular forms over ℚ, R(K_{0}, Y) is generated by the operators T_{𝔠}(𝔭) and S_{𝔠}(𝔭) for all prime ideal 𝔭.

By definition of the multiplication in R(K_{0}, Y) and as in the one dimensional case (see Theorem 3.24 [48, Chap.3, sec. 3]), we have

T_{𝔠}(𝔪)T_{𝔠}(𝔫) = ∑_{𝔪+𝔫⊂𝔞} N(𝔞)^{k−1}S_{𝔠}(𝔞)T_{𝔠}(𝔞^{−2}𝔪𝔫);

we also have the formal Euler product (see [47] and [48, Chap. 3, sec.3])

∑_{𝔞}T_{𝔠}(𝔞)N(𝔞)^{−s} = ∏_{𝔭}(1 − T_{𝔠}(𝔭)N(𝔭^{−s}) + S_{𝔠}(𝔭)N(𝔭)^{k−1−2s})^{−1} .

The Hecke operators act on the Fourier coefficients of f by

c(𝔪, T_{𝔠}(𝔞)f) = ∑_{𝔪+𝔫⊂𝔞} N(𝔞)^{k−1}c(𝔞^{−2}𝔪𝔫, S_{𝔠}(𝔞)f).

Since, the Hecke operators commute and are self-adjoint with respect to the Peterson inner product, they have a common basis of eigenvectors. If f is an eigenform, the corresponding eigenvalues are the c(𝔞, f). We say that f is normalized if c(O_{F} , f) = 1. When f is a normalized eigenform, its Dirichlet L–series has a formal Euler product

L(f, s) = ∏_{𝔭}(1 − c(𝔭, f)N(𝔭)^{−s} + N(𝔭)^{k−1−2s})^{−1}.

Proposition 1.31. The eigenvalues of T_{𝔠}(𝔞) are algebraic integers.

Proof. See [47, Prop. 2.2]. □

5. Automorphic representations.

We refer the reader to Jacquet–Langlands [32, Chap. I & II], Godement [26] or Gelbart [25, Chap. II & III], for the definition of admissible representations and the local and global Hecke theory; there is also a very nice survey of representations of p–adic groups by Cartier in [9]. We let H_{𝔭} be the local Hecke algebra at the prime 𝔭 and H_{𝔸} the global one (which is obtained as the restricted tensor product of the local ones; see Flath [23]).

Definition 1.32. Let Φ be in A^{k}(G, K_{f}). We say that W_{Φ} is an automorphic representation if it is irreducible. If further, Φ ∈ A_{0}^{k}(G, K_{f}), we say that W_{Φ} is cuspidal.

Remark 1.33. We only consider automorphic repesentations associated with the trivial character. By Theorem 4.5 [6], such a representation is admissible. Also, since we are only interested in unitary admissible representations, we will not distinguished between such representations and their corresponding H_{𝔸}–modules; indeed, the functor that sends the isomorphism class of a unitary admissible G(𝔸)–module to the one of the corresponding H_{𝔸}–module is full and faithful.

A natural question is the understanding of the map Φ ↦ W_{Φ}. This is very well–understood in the case of GL_{2}. We will restrict ourselves to this case in the rest of this section and refer to [25, Chap. 5, sec. A] and [32, Chap. 2, sec. 10], for more on that subject and for the following theorem which make use of the uniqueness of the Whittaker model.

5. AUTOMORPHIC REPRESENTATIONS. 27

Theorem 1.34 (Multiplicity one). The representation R_{0} decomposes into a countable direct (Hilbert) sum of cuspidal representations:

R_{0} = ⨁_{α} W_{α}.

In this decomposition, each W_{α} has multiplicity one (i.e., up to isomorphism, W_{α} appears only once). Moreover, for each α, β, if there is a finite set of places S such that for each 𝔭 ∉ S, W_{α, 𝔭} ≅ W_{β, 𝔭}, then W_{α} = W_{β}.

Take Φ ∈ A_{0}^{k}(G, K_{f}) and assume that the corresponding representation π_{Φ} is cuspidal. Then, by Theorem 4 in Flath [23], it factorizes as π_{Φ} = ⊗π_{𝔭}. The characterization of cuspidal representations is then purely local and we need the following few defintions to start.

Definition 1.35. Let π be an infinite dimensional irreducible admissible representation of G_{𝔭} = GL_{2}(F_{𝔭}) at a finite prime 𝔭 on the vector space V. We let π_{𝔭} be a uniformizer at 𝔭, and ω_{π} the central character of π. The conductor 𝔠_{π} of π is the largest ideal 𝔠 of O_{𝔭} such that V^{K_{1}(𝔠)} ≠ 0, where

K_{1}(𝔠) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(O_{𝔭})| c, d − 1 ∈ 𝔠\} .

We say that π is unramified (or class 1 or special) if 𝔠_{π} = O_{𝔭}.

Here is the list of possible conductors for a p-adic field (see Gelbart [25, Remark 4.25]):

• If π = π(χ_{1}, χ_{2}) is in the principal series, then 𝔠_{π} = 𝔣_{1}𝔣_{2}, where

𝔣_{i}, i = 1, 2 is the conductor of χ_{i}.

28 1. COMPUTING HILBERT MODULAR FORMS.

• If π = π(χ|·|^{1/2}, χ|·|^{−1/2}) is special, then 𝔠_{π} = 𝔣^{2} ∩πO_{𝔭}, where 𝔣 is the conductor of χ.

• If π is supercuspidal, then 𝔠_{π} = π^{n}O_{𝔭}, for some n ≥ 2.

Note that 𝔠_{π} is divisible by the conductor of ω_{π} and we have the following theorem.

Theorem 1.36. ([10, Theorem 1]) For 𝔠 = 𝔠_{π}, the space V^{K_{1}(𝔠)} is one-dimensional and

v ∈ V^{K_{1}(𝔠)} ⇔ π(g)v = ω_{π}(d)v, for all g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{0}(𝔠),

where

K_{0}(𝔠) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(O_{𝔭})|c ∈ 𝔠\} .

Definition 1.37. The vector v ∈ V, unique up to constant, invariant under K_{1}(𝔠), with 𝔠 = 𝔠_{π}, is called the new vector of π.

Piecing these together, we get the notion of a new form.

Definition 1.38. Let W be a cuspidal representation of weight k. Then, it is an irreducible admissible representation of G_{f}. Therefore, it factors as π = ⊗π_{𝔭}. For each finite prime 𝔭, let v_{𝔭} be a new vector of π_{𝔭}. We assume that, for almost all 𝔭, v_{𝔭} is equal to the vector we have used to define the restricted tensor product. We say that v = ⊗v_{𝔭}, defined up to a constant, is the new vector of W. Let Φ be in A_{0}^{k}(G, K_{f}), and W_{Φ} = Span(Φ · G_{f}). We say that Φ is a new form if W_{Φ} is irreducible and Φ is a new vector for W_{Φ}. The conductor of Φ is then the ideal 𝔠 = ∏_{𝔭} π_{𝔭}^{f_{𝔭}}, f_{𝔭} being the conductor of W_{Φ, 𝔭}.

6. THE JACQUET-LANGLANDS CORRESPONDENCE. 29

Remark 1.39. By Theorem 2 in Flath [23] and the discussion following it, almost every W_{𝔭} is unramified; therefore the new vector is welldefined up to a constant. Further we see that, by the multiplicity one theorem, W_{Φ}^{K_{1}(𝔠)} must be one-dimensional; which means that Φ is an eigenform.

We denote by A_{0}^{k, new}(G, K_{f}) the subspace of new forms in A_{0}^{k}(G, K_{f}).

Theorem 1.40. [16, Deligne, Scholie 2.4.2] The map Φ ↦ W_{Φ}, which associates to each cusp form the irreducible G_{f}-module it generates, is an injection of A_{0}^{k, new}(G, K_{f}) into the set of isomorphism classes of cuspidal representations.

When K_{f} = K_{0} = K_{0}(𝔠), the holomorphic newforms are just S_{k}^{new}(𝔠) = S_{k}(𝔠) ∩ A_{0}^{k, new}(G, K_{0}). For a more classical definition of newforms, see Shimura [47, sec. 2, Prop. 3].

6. The Jacquet-Langlands correspondence.

The last step in building the bridge between the new forms on the algebraic group G′ = Res_{F/ℚ}B^{×} and the ones on G = Res_{F/ℚ}GL_{2} is making explicit the correspondence between automorphic representations on G′_{𝔸} and on G_{𝔸}. Let Ram(B) be the set of places at which B is ramified. We take a cusp form π′ on B_{𝔸}^{×}, and let π′ = ⊗π′_{𝔭} be its factorization. If 𝔭 ∉ Ram(B), we fix an isomorphism B_{𝔭} = M_{2}(F_{𝔭}) such that the maximal compact open subgroup K′_{𝔭} we have fixed in defining the Hecke algebra of B goes to K_{𝔭} = GL_{2}(O_{𝔭}). We can carry the representation π′_{𝔭} to a representation π_{𝔭} of G_{𝔭} via this isomorphism,

30 1. COMPUTING HILBERT MODULAR FORMS.

and by construction, those two representations will be isomorphic as Hecke modules. Although our choice of isomorphism has been arbitrary, it will not affect the construction we will be making (see [32]). It remains now to take care of the local representations at places in Ram(B). This is done by making use of the Weil representations (see Jacquet-Langlands [32] and Gelbart [25], for more details). By taking the restricted tensor product of the local representations we just obtained, we get an admissible irreducible representation π = JL(π′). That π = JL(π′) is automorphic is a consequence of the celebrated Jacquet-Langlands correspondence.

Theorem 1.41. The map π′ ↦ π = JL(π′) is an injection of the irreducible constituents of A^{k}(G′, K′) into A_{0}^{k, new}(G, K). Its image consists of the representations π with π_{𝔭} square-integrable for 𝔭 ∈ Ram(B).

Proof. The proof of this result relies essentially on the trace formula and we refer to [25, Chap. 9 and 10] for a detailed treatment. □

7. Applications.

We would like to compute the coefficients a_{𝔭} (N(𝔭) ≤ 100) of all the (normalized) eigenvectors of the T_{𝔭} acting on S_{2}(𝔠) with N(𝔠) ≤ 500 on F = ℚ(\sqrt{5}). To this end, let us consider the (unique, up to isomorphism) totally definite quaternion algebra B over F, which is unramified at all finite places. The algebra B can be identified with the standard Hamilton quaternion algebra, since 2 is inert in F:

B = {x + yi + zj + wk, x, y, z, w ∈ F } .

7. APPLICATIONS. 31

As we have seen in Example 1.19, the class number of B, hence the type number is one. Every maximal order in B is then conjugate to the icosian ring

R = ℤ[ω][e_{1}, e_{2}, e_{3}, e_{4}],

whose group of units R^{×} is isomorphic to the semi-direct product of ℤ with the binary icosahedral group of order 120. (Cf. [13, Chap. 8, sec. 2.1.]) For any prime 𝔭, we fix an isomorphism B_{𝔭} = M_{2}(F_{𝔭}) so that R_{𝔭} = M_{2}(O_{F, 𝔭}). Now, let R_{𝔠} be an Eichler order of level 𝔠 in R, and write

R̂_{𝔠} = R_{𝔠} ⊗ Ô_{F} ,  R̂ = R ⊗ Ô_{F}  and B̂ = B ⊗ Ô_{F} .

We recall that R̂^{×}\B̂^{×}/B^{×} parametrizes the set of left ideal classes of R. Thus, since B has class number one,

R̂^{×}\B̂^{×}/B^{×} = {R̂^{×}B^{×}}, and B̂^{×}/B^{×} = R̂^{×}/R^{×}.

Hence

R̂_{𝔠}^{×}\B̂^{×}/B^{×} = R̂_{𝔠}^{×}\R̂^{×}/R^{×} = (∏_{𝔮|𝔠}R_{𝔠, 𝔮}^{×}\R_{𝔮}^{×}) /R^{×}
= (∏_{𝔮|𝔠}P^{1}(O_{𝔮}/𝔮^{e_{𝔮}})) /R^{×},

where 𝔠 = ∏_{𝔮|𝔠} 𝔮^{e_{𝔮}} and

P^{1}(A) = {(a, b) ∈ A^{2} : αa + βb = 1 for some (α, β) ∈ A^{2}} /A^{×},

for any ring A. This is then a finite set, and L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) is the vector space of ℂ-valued maps defined on it. By the Jacquet-Langlands correspondence, S_{2}(𝔠) is isomorphic as a Hecke module to the complex space

L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}),

32 1. COMPUTING HILBERT MODULAR FORMS.

on which the Hecke operators act in the standard way. To carry out our computations, we will explain a bit more how this action goes. This will lead us to the theory of Brandt matrices in the case of algebraic modular forms (as in Gross [28] and Khuri–Makdisi [33]). Indeed, in the spirit of Gross [28], L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) can be identified with the space of algebraic modular forms on G = Res_{F/ℚ}(B^{×}/F^{×}) of level the compact open subgroup with K = R̂_{𝔠}^{×} with values in the trivial representation V = ℂ. To make this more precise as a statement, let us recall some definitions and facts. We assume for a while that G is a connected reductive group defined over ℚ such that G(ℝ) is compact. We let V be a (finite) dimensional complex representation of G(ℝ) and fix a compact open subgroup K of G_{f} = G(𝔸_{f}).

Definition 1.42. [33] The space of algebraic modular forms of level K and weight V on G is M(V, K) := Hom_{G(ℝ)}(V, L^{2}(K\G_{𝔸}/G(ℚ))).

By Proposition 8.3 in [28] (see also [33, Lemma 2.2]), M(V, K) can be identified with the complex space of all functions f : K\G_{𝔸}/G(ℚ) → V^{∗} such that f(g_{∞}g) = g_{∞}^{−1}f(g) for all g_{∞} ∈ G(ℝ). This is in fact an isomorphism of Hecke modules.

Proposition 1.43. Let (g_{i})_{1≤i≤k} be a complete set of representatives of K\G_{f}/G(ℚ). Define Γ_{i} = G(ℚ)∩g_{i}^{−1}Kg_{i}. Then Γ_{i} is a finite group, as it embeds discretely into the compact group G(ℝ). Let V^{Γ_{i}} be the subspace of all Γ_{i}-invariant vectors in V. Then the map

M(V, K) → ⨁_{i=1}^{k} V^{Γ_{i}}
f ↦ (f(g_{i}))_{1≤i≤k}

is an isomorphism of complex spaces.

7. APPLICATIONS. 33

Proof. The proof essentially uses the fact that each element in M(V, K) viewed as a functions on the double coset space K\G_{𝔸}/G(ℚ) is completely determined by its values on the g_{i}; see [28, Proposition 4.5] or [33, Lemma 2.3]. □

By letting G = Res_{F/ℚ}(B^{×}/F^{×}) and K the image of R̂_{𝔠}^{×} by the projection B̂^{×} → G_{f}, and V = ℂ the trivial representation, we see that L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) is nothing else but the space of all algebraic modular forms on G of level K and trivial weight as we claimed before. The definition of Brandt matrices for the action of Hecke operators on such forms has been given by Khuri–Makdisi in the case when V is nontrivial. In what follows, we will content ourselves with recalling that definition only for algebraic modular forms of weight the trivial representation and refer to Khuri–Makdisi [33, sec. 3] for the general case. So, let (g_{i})_{1≤i≤k} be a complete set of representatives for the classes in R̂_{𝔠}^{×}\B̂^{×}/B^{×}, and recall that we have an isomorphism of complex spaces:

L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) → ℂ^{k}
f ↦ (f(g_{i}))_{1≤i≤k}.

Also recall that in this setting, the definition of Hecke operators is as follows: take t ∈ G_{f} and write the double coset as a disjoint union KtK = ∪_{i=1}^{s}Kt_{i}. Then,

f||KtK(g) = ∑_{i=1}^{s} f(t_{i}g), for g ∈ G_{𝔸}.

Viewing every element in L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) as a column vector via the isomorphism above, we see that the action of t on L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) is then given by a matrix B_{t}, called the Brandt matrix of t. We see that it is independent on both sets of coset representatives (g_{i})_{1≤i≤k} and

34 1. COMPUTING HILBERT MODULAR FORMS.

(t_{j})_{1≤j≤s}. For a prime 𝔭 ∈ O_{F}, we will simply denote by B_{𝔭} = (b_{ij}) the Brandt matrix of T_{𝔭}. When B is unramified at 𝔭 there is a well-known alternative description of the B_{𝔭} = (b_{ij}). Indeed, fix an isomorphism B_{𝔭} = M_{2}(F_{𝔭}) such that R_{𝔭} = M_{2}(O_{𝔭}), and let t_{𝔭} ∈ R_{𝔭} correspond

to (\begin{matrix}π_{𝔭} & 0 \\ 0 & 1 \end{matrix}), where π_{𝔭} is a uniformizer of O_{𝔭}. We can view t_{𝔭} as an

element of B̂^{×} by letting all its other entries be equal to the identity matrix in the corresponding B_{𝔮}’s. Then, the following proposition is standard.

Theorem 1.44. [37, Pizer] Let 𝔭 be a prime at which B splits such that (𝔭, 𝔠) = 1, and let I_{1}, …, I_{k} be a complete set of representatives of the ideal classes in R̂_{𝔠}^{×}\B̂^{×}/B^{×} so that I_{i} corresponds to g_{i}. Then b_{ij} is the number of subideals L_{ν} ⊂ I_{i} satisfying the following conditions:

(i) I_{i}/L_{ν} ≅ (O_{𝔭}/π_{𝔭}O_{𝔭})^{2}, (ii) L_{ν} is in the same ideal class as I_{j}; so that L_{ν} = γ_{ν}I_{j} for some γ_{ν} ∈ B^{×}.

The computation of the Brandt matrices exploits this theorem and the duality between the action of Hecke operators on L^{2}(R̂_{𝔠}^{×}\B̂^{×}/B^{×}) and on the free group on X = R̂_{𝔠}^{×}\B̂^{×}/B^{×}. Indeed, let us use the improper notation that Div(X) equals the free abelian group on X. We define the map

S_{2}(𝔠) × X → ℂ
(f, x) ↦ ⟨f, x⟩ = f(x), x ∈ X,

that we extend linearly in the second factor to Div(X)⊗_{ℤ}ℂ. One easily sees that this is a perfect pairing and by definition of Hecke operators,

⟨f||T, D⟩ = ⟨f, D||T ⟩,

for all f ∈ S_{2}(𝔠), D ∈ Div(X) and T ∈ R(K_{0}(𝔠), Y). Further, the action of Hecke on Div(X) ⊗_{ℤ} ℂ preserves the lattice Div(X) by definition. Therefore, it is enough to compute the action of Hecke on Div(X). The resulting matrices are then the tranposes of the Brandt matrices. The action of Hecke on Div(X) is described in several places. We refer to Gross [28, sec. 4] and Bertolini and Darmon [4, sec. 1.5] where this is done in the setting of quaternion algebras over ℚ. If B splits at the prime 𝔭 such that (𝔭, 𝔠) = 1, then

(R_{𝔠} ⊗ O_{𝔭})^{×}\B_{𝔭}^{×}/F_{𝔭}^{×} ≅ PGL_{2}(O_{𝔭})\PGL_{2}(F_{𝔭})

is the Bruhat-Tits tree of PGL_{2}(F_{𝔭}) whose vertices correspond to similarity classes of O_{𝔭}-lattices in F_{𝔭}^{2}. This is a homogeneous tree of degree N(𝔭) + 1. Let g ∈ PGL_{2}(O_{𝔭})\PGL_{2}(F_{𝔭}) corresponds to the lattice L. The neighbours of g which determine all sublattices of L of index 𝔭, are given by

g_{i} = (\begin{matrix}1 & i \\ 0 & π_{𝔭} \end{matrix}) g, i ∈ 𝔽_{𝔭}, g_{∞} = (\begin{matrix}π_{𝔭} & 0 \\ 0 & 1 \end{matrix}) g.

To get the action of T_{𝔭} on Div(X), we then take N(𝔭) + 1 global elements γ_{i}, i ∈ 𝔽_{𝔭} and γ_{∞} in R such that g_{i} = γ_{i}g, i ∈ 𝔽_{𝔭} and g_{∞} = γ_{∞}g. Then, by Theorem 1.44,

T_{𝔭}.x = ∑_{i∈𝔽_{𝔭}}γ_{i}.x + g_{∞}x, for x ∈ X.

36 1. COMPUTING HILBERT MODULAR FORMS.

When 𝔭|𝔠, the operator S_{𝔭} is described in a similar way. Practically, we let the elements γ_{i}, i ∈ 𝔽_{𝔭} and γ_{∞} act on the projective space ∏_{𝔮|𝔠} P^{1}(O_{𝔮}/𝔮^{e_{𝔮}}). We then take the summation over the elements which are not equivalent to (0, 0) under the action of R^{×}. To do the computations, we find a fundamental domain S of the action of R^{×} on ∏_{𝔮|𝔠} P^{1}(O_{𝔮}/𝔮^{e_{𝔮}}). We then let R act on S by embedding it diagonally into R̂.

Algorithms.

1) Generating the icosian group: We could have used a set of generators for the icosian group and their relations (cf. [13, Chap. 8, sec. 2.1] to generate this group. Instead, we have chosen to find all 4–tuples in ℤ[ω] such that the quaternion q = x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4} has reduced norm 1 (see example 1.19).

2) Finding a fundamental domain: We first find a set of representatives of the space P^{1}(O/𝔠). We have chosen to work with the product

P^{1}(O/𝔠) = ∏_{𝔭|𝔠}P^{1}(O/𝔭^{e_{𝔭}}).

Then the coset representatives for each local factor P^{1}(O/𝔭^{e_{𝔭}}) are taken to be all couples

(1, a) a ∈ 𝔭/𝔭^{e_{𝔭}} and (a, 1) a ∈ O/𝔭^{e_{𝔭}}.

We let the icosian group R^{×} acts on this set of coset representatives. This is done via the local isomorphism R_{𝔭} = M_{2}(O_{𝔭}) which reduces to

R ⊗ (O_{F} /𝔭^{e_{𝔭}}) = M_{2}(O_{F} /𝔭^{e_{𝔭}}).

7. APPLICATIONS. 37

Note that by Hensel’s Lemma, we only need find that reduced isomorphism. This amounts to finding a set of generators of M_{2}(O_{F} /𝔭^{e_{𝔭}}) which satisfies the appropriate relations corresponding to the basis we have chosen for R_{𝔭}. This determines a fundamental domain for this action.

3) Generating the Hecke operators: Let 𝔭 be a prime in F = ℚ(\sqrt{5}). To compute T_{𝔭}, it is enough to find coset representatives of reduced norm (a uniformizer of) 𝔭 in the form

q = x + yi + zj + wk with x, y, z, w ∈ ℤ[ω].

To this end, we again consider the local isomorphism R_{𝔭} = M_{2}(O_{𝔭}) and reduce it to R ⊗ 𝔽_{𝔭} = M_{2}(𝔽_{𝔭}). We then consider the Eichler order in M_{2}(O_{F}) whose reduction consists of all upper triangular matrices. We use the fact that GL_{2}(O_{𝔭}) acts transitively on P^{1}(𝔽_{𝔭}) and that the stabilizer of ∞ under this action is the group of invertible elements in our fixed Eichler order. Our algorithm consists in finding global elements q ∈ R of reduced norm 𝔭 such that q′q^{−1} ∈ R_{𝔭}^{×} reduces to a non upper triangular element. This could have been much easier and faster if we were able to prove that R is Euclidean. Which we couldn’t do, so we have chosen instead to represents the quadratic form n(q) = x^{2} + y^{2} + z^{2} + w^{2}; so our algorithm is O(N(𝔭)^{2}) time. If p ∈ ℤ is a prime that splits, we only stored the cosets representatives corresponding to T_{𝔭} since we can get T𝔭̄ by Galois conjugation.

Conventions:

• Let p be a prime in ℤ which splits in O_{F} = ℤ[ω]. We have a canonical isomorphism O_{F} = ℤ[x]/(x^{2} − x − 1), and

O_{F} ⊗ 𝔽_{p} = 𝔽_{p}[x]/(x^{2} − x − 1) = 𝔽[x]/(x − α) × 𝔽[x]/(x − β),

where α is the reduction of ω modulo p and β the one of ω̄. We denote by pA the prime which reduces to (x − α), and pB the one which reduces to (x − β).

• For quadratic fields, we use the notations

ω_{D} = \{ \begin{matrix}\sqrt{D} & if D ≠ 1 mod 4 \\ \frac{1+\sqrt{D}}{2} & if D = 1 mod 4. \end{matrix}

• The prime 𝔭 has norm 2 ≤ N(𝔭) ≤ 100.

• The ideal of smallest norm above which we could find a modular form has norm 31. The level 𝔠 has norm 31 ≤ N(𝔠) ≤ 500.

• The first row of the table gives the levels which have been listed in the increasing order of the norm. For each level 𝔠, the corresponding columns give the list of all modular forms of level 𝔠. The listing is done up to Galois conjugation of the levels.

• We have listed separately all levels 𝔠 which have normalized eigenforms with Fourier coefficients generating a field of dimension greater or equal to 3. In such case, the notation 𝔠 : P_{1}, …, P_{s} means that, up to Galois conjugation, we have s newforms f_{1}, …, f_{s} and that the field generated by the Hecke eigenvalues of f_{i} is ℚ(α_{i}), where α_{i} is a root of the irreducible polynomial P_{i} which is also an Hecke eigenvalue. We then

7. APPLICATIONS. 39

give the coefficients of each eigenform in a chosen integral basis. We only specify this integral basis when it does not correspond to the canonical power basis {1, α_{i}, …, α_{i}^{d_{i}−1}}, where d_{i} = deg P_{i}.

31A  2.3 41A 3.\sqrt{5}  7 \sqrt{5}.11A  61A  2^{3} 71A
𝔭 a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f)  a_{𝔭}(f)  a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f)
2  −3  −  −2  −3  0  −1  −2ω_{5}  −  −1
3  2  −  −4  −  5  −2  −3 + ω_{5}  2  −2
\sqrt{5}  −2  −4  −1  −  −4  − −2 + 3ω_{5}  −2  0
7  2  10  −6 −14  −  14 −2 + 4ω_{5}  10  −4
11A  4  2  −2  −4  −3  −  −1 + ω_{5}  −4  0
11B  −4  2  5  −4  −3  0  2 − 4ω_{5}  −4  0
19A  −4  0  −1  4  0  8  2 − ω_{5}  4  −4
19B  4  0  6  4  0  −4 −3 − 3ω_{5}  4  2
29A  −2  0  9  −2  5  −6 −4 + 5ω_{5}  −2  −6
29B  −2  0  2  −2  5  6  4 + 2ω_{5}  −2  6
31A  −  −8 −10  0  2  −4  4 − 5ω_{5}  0  2
31B  8  −8  4  0  2  8  6 + 2ω_{5}  0  8
41A  −6  2  −  10  2  −6  6 − 2ω_{5}  2  12
41B  −6  2  0  10  2  6 −10 + 4ω_{5}  2  6
59A  12 −10  −3  −4 −10  0 4 − 10ω_{5}  12 −12
59B  −4 −10  4  −4 −10  −12  7ω_{5}  12  6
61A  6  2  6  −2  −8  2  − −10  −4
61B  −2  2  −8  −2  −8  −10  0 −10 −10
71A  0  12  9  −8  −8  0  2 + 3ω_{5}  8  −
71B  −8  12 −12  −8  −8  0  −4ω_{5}  8  6
79A  0  0 −11  0  5  8 −8 + 2ω_{5} −16  14
79B  16  0  −4  −8  5  −4  8 − 4ω_{5} −16  −4
89A  −6  10  −8  −6  0  −18  2 + 2ω_{5}  −6  18
89B  10  10  −1  −6  0  6 −11 + 3ω_{5}  −6  6

2 · 19A  79A 2^{2}.\sqrt{5}  3^{2} 89A \sqrt{5}.19A 3.11A
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f)  a_{𝔭}(f) a_{𝔭}(f)
2  −  −  −  1  −  −1  −1  −1  1
3  −5  1  1  −2  −2  −  4  −2  −
\sqrt{5}  1  −3  −3  −2  −  0  0  −  −2
7  0  −4  −4  −2 −10  14  −4  2  2
11A  2  −6  3  0  0  0  0  0  −
11B  −3  3  −6  −4  0  0  −6  0  −4
19A  −  −  −  8  −4  −4  −4  −  4
19B  5  −7  −7  4  −4  −4  2  −4  −4
29A  5  3  −6  −2  6  0  6  6  6
29B  −10  −6  3  6  6  0  6  −6  −2
31A  −3  5  5  −8  −4  8  −4  −4  8
31B  7  5  5  0  −4  8  −4  8  −8
41A  2  6  6  −2  6  0  0  −6  −6
41B  2  6  6  2  6  0  6  −6  2
59A  10  6 −12  −4  12  0  0  12  12
59B  0 −12  6  4  12  0  12  12  12
61A  12  8  8  14  2  2  −4  −10  −2
61B  −8  8  8  10  2  2  14  14  −2
71A  −8  0  −9  12 −12  0  0  0  8
71B  7  −9  0 −16 −12  2  12  12  −8
79A  15  −1  −1  −  8 −16  2  −16  8
79B  5  −1  −1  8  8 −16 −16  8  16
89A  0  0  9  18  −6  0  −  −6  2
89B  −15  9  0 −14  −6  0  6  6 −14

2 · \sqrt{5}^{2}  101A  109A  2.29A  11A^{2}  11
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2})  a_{𝔭}(f)  a_{𝔭}(f) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f) a_{𝔭}(f_{1})  a_{𝔭}(f_{2})
2  −  −  −1 − ω_{2}  2ω_{5}  −  − −ω_{3}  0  1 − ω_{33}
3  −5  5 −2 − 2ω_{2}  1 − 3ω_{5}  0  −4  −1  −5  2 + ω_{33}
\sqrt{5}  −  −  1 + 2ω_{2}  −2ω_{5}  1  −1  ω_{3}  1 −2 + ω_{33}
7  −10  10  4 − 2ω_{2}  2 − 5ω_{5}  5  1  ω_{3}  −10  2
11A  −3  −3  −2ω_{2} −2 − ω_{5}  2  −2  −  −  −1
11B  −3  −3  1 + ω_{2} −4 + 2ω_{5}  −3  5  2ω_{3}  −  −1
19A  5  −5 −1 + 5ω_{2}  4 + 2ω_{5}  −5  −1  −2  0  −4
19B  5  −5  −3 − ω_{2}  −3ω_{5}  0  −8 −4ω_{3}  0  −4
29A  0  0  1 − 2ω_{2} −4 + 7ω_{5}  −  −  5ω_{3}  0 −2 − 2ω_{33}
29B  0  0  −4  5 − ω_{5}  −5  9  −3  0 −2 − 2ω_{33}
31A  2  2  −2 1 − 3ω_{5}  7  −3 −2ω_{3}  7  ω_{33}
31B  2  2 −3 − 3ω_{2} −3 + 3ω_{5}  −8  4 −2ω_{3}  7  ω_{33}
41A  −3  −3  1 − 4ω_{2}  6 − ω_{5}  −3  −7  9  −8 2 + 2ω_{33}
41B  −3  −3  8 −6 + 6ω_{5}  12  0 −4ω_{3}  −8 2 + 2ω_{33}
59A  0  0  2 − 4ω_{2}  1 − 5ω_{5}  0  4  −6  5  4 + ω_{33}
59B  0  0  4 + 2ω_{2} −4 + 8ω_{5}  0  4  −6  5  4 + ω_{33}
61A  2  2  11  −6ω_{5}  12  −8  3ω_{3}  12 6 − 2ω_{33}
61B  2  2  6 + 2ω_{2}  −6ω_{5}  2  6  7  12 6 − 2ω_{33}
71A  12  12 −6 + 4ω_{2}  4 − 8ω_{5}  7  9  6  −3  −3ω_{33}
71B  12  12  2 − 6ω_{2} −4 + 4ω_{5}  −8  16  6  −3  −3ω_{33}
79A  −10  10  −2ω_{2}  5 − 3ω_{5}  −10  10  6ω_{3}  −10 −8 + 2ω_{33}
79B  −10  10  2 + 4ω_{2}  8 − 2ω_{5}  −6 −11  14  −10 −8 + 2ω_{33}
89A  15 −15  7 + 6ω_{2}  3 − 7ω_{5}  15  −1  4ω_{3}  15  2 − ω_{33}
89B  15 −15 −10 + 6ω_{2}  16 + ω_{5}  0  −8 −7ω_{3}  15  2 − ω_{33}

131A 2.31A  \sqrt{5}^{3}  139A 2^{2}.3  \sqrt{5}.29A  149A
𝔭  a_{𝔭}(f_{1}) a_{𝔭}(f)  a_{𝔭}(f)  a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f)
2  −ω_{3}  −  0  1 − ω_{2}  −  −2  1  2  0
3 −2 + 2ω_{3}  −2  0 −1 − ω_{2}  −  3  −6  1  2 − 4ω_{5}
\sqrt{5}  ω_{3}  0  −  2ω_{2}  2  −  −  −  2ω_{5}
7 −2 − 6ω_{3}  14  0 6 − 4ω_{2}  −2  8  2  −4  2 + ω_{5}
11A  2 − 2ω_{3}  −6 2 − 5ω_{5} −3 − 2ω_{2}  −4  −2  4  −6  −ω_{5}
11B  1  0 −3 + 5ω_{5}  −2  −4  5  −4  −3 −2 − 2ω_{5}
19A  −2  2  0 1 + 4ω_{2}  0  −8  4  −4  3 − ω_{5}
19B  −2  2  0  3 + ω_{2}  0  −1  −4  5 −5 + 5ω_{5}
29A −4 + 2ω_{3}  0  0 −3 + 4ω_{2}  6  −  −  −  5ω_{5}
29B −1 − 4ω_{3}  6  0 −1 − 6ω_{2}  6  −5  −2  9  7 + ω_{5}
31A −3 + 4ω_{3}  − −8 + 5ω_{5}  7 − ω_{2}  4  −10  8  2 −6 + 6ω_{5}
31B  4 + ω_{3}  −4 −3 − 5ω_{5} −2 + 2ω_{2}  4  −3  0  5  −6ω_{5}
41A  4 − ω_{3}  6 7 − 5ω_{5} −1 + 5ω_{2} −10  7  10  −9 −2 − 2ω_{5}
41B  −4ω_{3}  −6 2 + 5ω_{5} −8 − 2ω_{2} −10  0  −6  0 −3 + 5ω_{5}
59A  −ω_{3}  −12  0 6 + 4ω_{2}  −4  −3 −12  −9  4 − 8ω_{5}
59B  6 − 2ω_{3}  0  0 10 − 2ω_{2}  −4  4  4  12  2 + ω_{5}
61A  6 + 2ω_{3}  8 2 − 5ω_{5} −3 + 3ω_{2}  2  13  −2  5  3 + 3ω_{5}
61B  −4 −10 −3 + 5ω_{5}  2 − 8ω_{2}  2  −8  −2  −4 −2 − 7ω_{5}
71A  4 + 4ω_{3}  −12 7 + 5ω_{5} −2 + 6ω_{2}  0  −12  0  12 −8 + 10ω_{5}
71B  4 + 4ω_{3}  0 12 − 5ω_{5} −2 − 8ω_{2}  0  2 −16  −6  6 − 3ω_{5}
79A  1 − 8ω_{3}  8  0 −2 + 4ω_{2}  12  10  16  2  2 − 9ω_{5}
79B  4 − 2ω_{3}  8  0 2 − 2ω_{2}  12  −4  8  −4  2 + ω_{5}
89A −7 − 4ω_{3}  6  0  6  10  6  −6  −6 12 − 9ω_{5}
89B  13  −6  0 4 − 4ω_{2}  10  13  10  15  6 − 2ω_{5}

\sqrt{5}.31A 2.41A  3.19A  2^{2}.11A  179A  2.3.\sqrt{5}
𝔭  a_{𝔭}(f) a_{𝔭}(f) a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f)
2  1  −  −1 −1 + 2ω_{5}  −  −  −1  1  −
3  −2  2  −  −  −2 2 − 4ω_{5}  4  0  −
\sqrt{5}  −  2  0  2 − 2ω_{5}  0  2ω_{5}  0  0  −
7  −6  −6  −10  2 − 4ω_{5}  8 −8 + 6ω_{5}  5  −5  2
11A  0  −2  0  4 − 4ω_{5}  −  −  3  −5  0
11B  4  −4  6 −4 + 2ω_{5}  6 −2 − 2ω_{5}  −3  3  0
19A  0  8  −  −  −4  0  8  −4  −4
19B  −4  −6  −4  0  2 −6 + 2ω_{5}  −1  1  −4
29A  −2  −6  −6 −2 + 4ω_{5}  −6 2 − 4ω_{5}  −3  5  −6
29B  −2  8  0 −6 + 2ω_{5}  −6 −2 + 4ω_{5}  −9  7  −6
31A  −  −4  −4  4 − 4ω_{5}  2 −6 + 6ω_{5}  −10  2  8
31B  −4  −8  −4  4ω_{5}  −4 4 − 4ω_{5}  −7  1  8
41A  2  −  −6 −2 + 8ω_{5}  −6 −2 + 8ω_{5}  3  −3  −6
41B  2  6  12 −2 − 2ω_{5}  0 8 − 2ω_{5}  −6  6  −6
59A  −12  0  6 −8 + 6ω_{5}  12 −4 + 8ω_{5}  0  8  0
59B  −12  4  −6 −4 − 2ω_{5}  0 −8 − 4ω_{5}  12  −4  0
61A  10  0  14  2  8  −6ω_{5}  2 −10  −10
61B  −2  −8  2 −10 + 4ω_{5}  −10  2  −13 −11  −10
71A  −4  0  −6  4 + 6ω_{5}  0  12  6  10  0
71B  16  0  6  8 − 2ω_{5}  12 4 − 4ω_{5}  0  12  0
79A  −8  16  8  4 − 8ω_{5}  −10 −2 − 6ω_{5}  8  12  8
79B  16  −4  −4  8 + 4ω_{5}  8  0  8  0  8
89A  10  10  6  2 − 4ω_{5}  6 −2 + 4ω_{5}  −12  4  18
89B  18  14  12 −2 − 6ω_{5}  6 6 − 12ω_{5}  0  4  18

191A  2.7  199A
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  −3  1  −  −  0  −4  3
3  2 2 − 4ω_{5}  −2  0  0  −2  0
\sqrt{5}  2 −2 + 2ω_{5}  0  2 − \sqrt{6}  1  −3  −2
7  −2 −6 + 2ω_{5}  −  −  5  5  2
11A  0 4 − 2ω_{5}  0  2\sqrt{6}  2  0  −4
11B  4  0  0  2\sqrt{6}  −3  −3  0
19A  −4 −4 + 4ω_{5}  2 −4 − \sqrt{6}  0  2  0
19B  4  −4  2 −4 − \sqrt{6}  5  −7  −4
29A  −2 −6 + 4ω_{5}  −6 −2 + 2\sqrt{6}  0  6  −6
29B  −10 6 − 4ω_{5}  −6 −2 + 2\sqrt{6}  0  −6  6
31A  0  0  −4 4 − 2\sqrt{6}  −8  −4  10
31B  −8  4ω_{5}  −4 4 − 2\sqrt{6}  2  −4  −4
41A  2 −6 + 4ω_{5}  6 −6 − 2\sqrt{6}  −3  −3  6
41B  −10 2 + 2ω_{5}  6 −6 − 2\sqrt{6}  −8  12  4
59A  0 4 − 10ω_{5}  −6  4 + \sqrt{6}  −5  3  4
59B  12 −4 + 8ω_{5}  −6  4 + \sqrt{6}  10  0  −2
61A  10 6 − 6ω_{5}  8  6 + \sqrt{6}  2 −10  14
61B  14 −2 + 8ω_{5}  8  6 + \sqrt{6}  7  −1 −14
71A  12 8 − 10ω_{5}  0 −6 − 2\sqrt{6}  2 −12  8
71B  0 4 − 8ω_{5}  0 −6 − 2\sqrt{6}  −3  −3 −12
79A  −16  0  8 −2 + 2\sqrt{6}  10 −10  10
79B  −4  2ω_{5}  8 −2 + 2\sqrt{6}  −10 −10  8
89A  −14 10 + 4ω_{5}  −6  10  −10  0  14
89B  −6  −6  −6  10  0  12  6

\sqrt{5}.41A  11A.19A  11B.19A
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  −1  1  3  −3  −1  1  −2  1  2
3  4  2  −4  −4  4  −2  2  2  −2
\sqrt{5}  −  −  −  −4  0  2  4  −2  0
7  −4  10  4  −12  8  6  −12  −6  8
11A  6  −4  −2  −  −  −  −5  4  −3
11B  0  −4  0  0  0  −4  −  −  −
19A  −4  −4  4  −  −  −  −  −  −
19B  −4  4  −4  2  2  0  1  4  −7
29A  −6  −2  −6  2  −6  6  1  −2  9
29B  0  6  −8  −10  6  −2  7  −2  9
31A  12  0  10  4  −4  0  −5  −8  5
31B  2  0  −6  −8  −4  0  −2  −8 −10
41A  −  −  −  0  0  6  −3  −6  3
41B  −6  −6  10  0 −12  6  −11  10  −3
59A  −12  4  12  4 −12  4  −3  12  −3
59B  12 −12  4  −10  6  8  10  4  −6
61A  −10  −2  6  6  2  −2  8 −10  −4
61B  2  14  2  −4  −4  −6  −2  −2  2
71A  −12  8  4  −8  0  0  −13  8  3
71B  6  −8  −2  −14  −6 −12  12  0  0
79A  8  −8 −16  −8 −16  8  4  −8  8
79B  −4  0  −4  16  8 −16  −10  8  2
89A  0  18  −8  14  6  18  10  10  6
89B  −6  −6  −6  −6  6  10  −14  10  6

2.\sqrt{5}.11A  3.\sqrt{5}^{2}
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  −  −  −  0  0  3
3  0  −2  4  −  −  −
\sqrt{5}  −  −  −  −  −  −
7  −10  2  2  −5  5  14
11A  −  −  −  2  2  −4
11B  2  0  −6  2  2  −4
19A  0  −4  −4  −5  5  −4
19B  0  −4  8  −5  5  −4
29A  −10  6  −6  10 −10  2
29B  0  −6  0  10 −10  2
31A  −8  8  −4  −3  −3  0
31B  2  −4 −10  −3  −3  0
41A  2  −6  −6  −8  −8  10
41B  −8  6  0  −8  −8  10
59A  0 −12  12  −10  10  4
59B  10  0  −6  −10  10  4
61A  2 −10  14  7  7  −2
61B  2  14  2  7  7  −2
71A  12  0 −12  −8  −8  −8
71B  −8  0  0  −8  −8  −8
79A  0  8  8  0  0  0
79B  10  −4 −10  0  0  0
89A  10  −6  6  0  0  6
89B  −10  −6  −6  0  0  6

2 · 59B  239A
𝔭  a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  1  −1  −3  1  1
3  −2ω_{5}  2ω_{21}  −5  −1  2
\sqrt{5}  1 − ω_{5}  1 − ω_{21}  −3  1  −2
7  −8  −4ω_{21}  6  −6  6
11A −2 + 3ω_{5}  2 − ω_{21}  1  −5  4
11B −2 + 3ω_{5}  2 + ω_{21}  −3  3  0
19A  −4ω_{5}  0  7  5  −4
19B −3 + 7ω_{5} −5 + ω_{21}  −2  −2  4
29A  5 − 5ω_{5} −5 + ω_{21}  −8  0  6
29B −4 − ω_{5}  −3ω_{21}  −8  0  6
31A  2 − 6ω_{5} 2 − 2ω_{21}  4  −4  −4
31B −6 + 4ω_{5}  2  3  1  −8
41A  7 − 7ω_{5} −3 + 3ω_{21}  3  1  −8
41B  8 − ω_{5} −8 + ω_{21}  −11  9  −6
59A −3 − ω_{5} 5 + 3ω_{21}  −7  11  −4
59B  1  −1  10  14  −4
61A −6 − 2ω_{5} 2 + 2ω_{21}  −6 −14  −2
61B  5ω_{5} −8 + ω_{21}  −7  5 −10
71A −1 − 3ω_{5} −3 + 3ω_{21}  14  −6  0
71B  14 2 − 4ω_{21}  −12  −8  16
79A −2 + ω_{5} −10 + ω_{21}  −12  −8  16
79B  2 − 4ω_{5} 10 − 4ω_{21}  −1  −3  0
89A −1 + 5ω_{5} −5 + 5ω_{21}  0  −4  2
89B  4 + 6ω_{5}  −6ω_{21}  −7  9  −6

2.61A  \sqrt{5}.7  2^{4}  269A
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f) a_{𝔭}(f_{1}) a_{𝔭}(f_{2})  a_{𝔭}(f_{3})
2  −  −  −4  ω_{17}  −  −3  −1  2
3  1  5  −5 −1 − ω_{17}  −2  −1  1  −2ω_{13}
\sqrt{5}  0  0  −  1  2  −3  3  0
7  −4  8  −  1 −10  −5  5 −2 + 4ω_{13}
11A  0  −4  −3  1 − ω_{17}  −4  0  0 −2 + 2ω_{13}
11B  −3  3  −3  1 − ω_{17}  4  −5  −3  0
19A  2  −6  2 −2 − 2ω_{17}  4  −3  5 −5 + ω_{13}
19B  2  6  2 −2 − 2ω_{17}  −4  1  −1 −2 + 4ω_{13}
29A  0  −8  3 −1 + 3ω_{17}  −2  −5  3 6 − 3ω_{13}
29B  −9  9  3 −1 + 3ω_{17}  −2  −2  −6 4 + 2ω_{13}
31A  5  −3  −4  0  0  6 −10 4 − 5ω_{13}
31B  −4  0  −4  0  0  −10  2 2 − 2ω_{13}
41A  −6 −10  −12  2ω_{17}  −2  12  0 5 − 5ω_{13}
41B  0  0  −12  2ω_{17}  −2  6  6 −8 − ω_{13}
59A  3  1  0  −4  12  3  3 6 + 3ω_{13}
59B  6 −10  0  −4 −12  −7  −9 10 − 4ω_{13}
61A  −  −  8  6ω_{17}  10  6  2  −4
61B  −1 −11  8  6ω_{17}  10  2 −10  2ω_{13}
71A  −3  −7  0  8  8  10  −6  6ω_{13}
71B  6 −10  0  8  −8  −6  −6 −7 + ω_{13}
79A  8  8  −1 −5 + ω_{17}  16  12 −16  1 + ω_{13}
79B  −1 −13  −1 −5 + ω_{17} −16  −8  8 −6 − ω_{13}
89A  12  −4  −12 4 − 2ω_{17}  −6  −18  6  7 − ω_{13}
89B  −3  13  −12 4 − 2ω_{17}  −6  −13  3 3 + 3ω_{13}

5 · 11A  3 · 31A  2 · 71A
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2})
2  1 1 − 2ω_{5}  −1  1  3  −  −
3  2 −2 + 4ω_{5}  −  −  −  2 2 − 2\sqrt{2}
\sqrt{5}  −  −  2  2  −2  0  2\sqrt{2}
7  −14 2 − 4ω_{5}  −2  10 −10  −2  5\sqrt{2}
11A  −  −  −2  0  −2  −2  −3\sqrt{2}
11B  0  2  2  −4  2  4  −2
19A  −8 −2 + 4ω_{5}  0  8  −4  6 −4 + \sqrt{2}
19B  4  0  4  −4  4  2 −2 − 2\sqrt{2}
29A  6 −2 + 4ω_{5}  0 −10  4  4  6 + \sqrt{2}
29B  −6 4 − 8ω_{5}  6  −2  −2  0 −2 − \sqrt{2}
31A  −4  2  −  −  −  −8 −6 + \sqrt{2}
31B  8  −8  −8  4  −4  8 −2 + 2\sqrt{2}
41A  −6  2  −12  10 −12  −6  −7\sqrt{2}
41B  6  −8  0  −6  0  −10 2 + 4\sqrt{2}
59A  0 −2 + 4ω_{5}  −4  4  12  0 12 + 2\sqrt{2}
59B  12 4 − 8ω_{5}  10  4  2  2 6 + 4\sqrt{2}
61A  2  −8  0  −6  0  8 −4 + 8\sqrt{2}
61B  −10  2  −6  14  10  −10  2\sqrt{2}
71A  0  −8  10  −4  −6  −  −
71B  0  12  8  0  −8  8 −6 + 5\sqrt{2}
79A  −8 4 − 8ω_{5}  −8  8  0  4 −6 + 3\sqrt{2}
79B  4 2 − 4ω_{5}  −8  0  −8  12  −4
89A  18 2 − 4ω_{5}  −6  10  18  −10 −4 + 8\sqrt{2}
89B  −6 −2 + 4ω_{5}  12  −6  −8  −6 4 − 4\sqrt{2}

2^{2}.19A  \sqrt{5}.61A  2^{3}.\sqrt{5}
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  −  −  1  1 − 2ω_{5}  −  −  −
3  4  2 − 2ω_{5}  2  2  2  2  −6
\sqrt{5}  −2  2  −  −  −  −  −
7  −6  6  2  2 − 4ω_{5}  −6  −6  2
11A  0 −4 + 4ω_{5}  −4  4  −4  4  4
11B  2  −2ω_{5}  0 −2 + 4ω_{5}  4  −4  4
19A  −  −  0 −6 + 4ω_{5}  −4  −4  4
19B  8 −4 + 4ω_{5}  0 −2 + 4ω_{5}  −4  −4  4
29A  2  2 + 4ω_{5}  6  2  −10  6  −2
29B  −10  −2  −2 −2 − 4ω_{5}  6 −10  −2
31A  −2 −4 − 2ω_{5}  −8  4 − 8ω_{5}  8  0  −8
31B  −10  −2  −4 −6 + 4ω_{5}  0  8  −8
41A  −4  2 − 2ω_{5}  2  2 − 8ω_{5}  −6  10  −6
41B  −10 −2 − 4ω_{5}  2  10  10  −6  −6
59A  −12  8  4  −8  −12  4  −4
59B  10 −8 − 2ω_{5}  0 −6 + 4ω_{5}  4 −12  −4
61A  4 −2 + 10ω_{5}  −  −  −2  −2  −2
61B  0 −10 + 6ω_{5}  −2 −6 + 8ω_{5}  −2  −2  −2
71A  −8 −8 + 8ω_{5}  0  4 − 8ω_{5}  −8  0  0
71B  −2 −4 − 2ω_{5}  −4  2 − 4ω_{5}  0  −8  0
79A  14 4 − 10ω_{5}  8 −4 + 8ω_{5}  −16  0  0
79B  0  8 − 8ω_{5}  −4 −2 + 12ω_{5}  0 −16  0
89A  6  6 − 4ω_{5}  10 −10 + 4ω_{5}  10  10  −6
89B  6 −6 + 12ω_{5}  −6 10 − 8ω_{5}  10  10  −6

2 · 79B  319A
𝔭  a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{4}) a_{𝔭}(f_{5}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2})  a_{𝔭}(f_{3})
2  1  −1  1  1  1  −3  1 − \sqrt{2}  2 − \sqrt{3}
3  3 − ω_{17}  1 − \sqrt{6}  2  −3  1  −2 2 − 2\sqrt{2} −2 + 2\sqrt{3}
\sqrt{5}  −1 − ω_{17}  1  −2  2  0  −2  2\sqrt{2}  0
7  −2 −8 − 2\sqrt{6}  −2  10  2  −10 −2 − 6\sqrt{2}  8
11A  2ω_{17}  2  2  0  0  −1  1  −1
11B  2ω_{17}  2\sqrt{6}  2  0  0  −4  −2  2\sqrt{3}
19A  1 + ω_{17}  1 − \sqrt{6}  2  −5  −1  −4  2\sqrt{2} 2 − 2\sqrt{3}
19B  −6  2 − 2\sqrt{6}  −6  0  −4  0  2\sqrt{2}  2
29A  −3 − ω_{17} −3 + 3\sqrt{6}  −4  −3  −3  −2 −4 + 2\sqrt{2}  4\sqrt{3}
29B  2 − 4ω_{17} −2 + 2\sqrt{6}  −2  −4  0  1  1  −1
31A  4 − 4ω_{17}  2\sqrt{6}  0  −2  2  −4 −4 − 2\sqrt{2}  2
31B  1 + 3ω_{17}  1 − 4\sqrt{6}  4  4  2  8 2 − 4\sqrt{2} −4 + 2\sqrt{3}
41A  −10  2 −10  2  −6  −2 6 − 2\sqrt{2} −6 − 2\sqrt{3}
41B  −3 + ω_{17}  1 + \sqrt{6}  −2  −3  −3  6 −6 + 2\sqrt{2}  −4\sqrt{3}
59A  3 + 3ω_{17} −11 + \sqrt{6}  6  3  3  4  6\sqrt{2} −6 + 2\sqrt{3}
59B  1 − ω_{17} −7 + 2\sqrt{6}  0  −2  0  0  4 6 − 4\sqrt{3}
61A  8 − 4ω_{17}  −8  4  8  8  −10  2\sqrt{2} −4 − 4\sqrt{3}
61B  3 − ω_{17} −1 − 2\sqrt{6}  2  0  2  6 −6 + 4\sqrt{2} 2 + 4\sqrt{3}
71A −1 − 3ω_{17}  5 + 2\sqrt{6}  −4  2  0  8 −8 − 6\sqrt{2} −6 + 2\sqrt{3}
71B −4 + 4ω_{17}  6 − 4\sqrt{6}  0 −10  −6  12 4 + 4\sqrt{2} −6 − 4\sqrt{3}
79A −10 + 2ω_{17}  2 − 2\sqrt{6}  −8 −10 −10  −8  10 −4 − 2\sqrt{3}
79B  1  −1  1  1  1  −8  8\sqrt{2} −4 + 4\sqrt{3}
89A  −14  10 −14  10  −6  10 −4 − 6\sqrt{2}  −8\sqrt{3}
89B  2ω_{17} −12 + 2\sqrt{6}  2  0  0  −14 −6 + 4\sqrt{2} 6 + 4\sqrt{3}

2.3^{2}  2.\sqrt{5}.19A  2.3.11A
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{4}) a_{𝔭}(f_{5})
2  −  −  −  −  −  −  −  −  −  −
3  −  −  2  −2  2  −  −  −  −  −
\sqrt{5}  4  0  −  −  −  −2  3  1  1  3
7  10  −4  0  −4  −2  −14 −11  −5  5  11
11A  −2 −3\sqrt{2}  4  0  −4  −  −  −  −  −
11B  −2 3\sqrt{2}  2  6  0  4  2  2  2  −6
19A  0  2  −  −  −  4  7  −5  −5  −1
19B  0  2  −2  2  0  4  −7  −5  5  −1
29A  0 6\sqrt{2}  4  0  2  −10  0  0  0  0
29B  0 −6\sqrt{2}  −2  −6  6  6  −7  5  −5  −9
31A  −8  −4  0  −4  −8  0  6  2  2 −10
31B  −8  −4  4  −4  −8  0  −1  −3  7  5
41A  −2 −3\sqrt{2}  −12  12 −10  −6  −5  7  −3  9
41B  −2 3\sqrt{2}  2  −6  10  −6  2  2  2  −6
59A  10 −3\sqrt{2}  0  0  −4  −12  4  0  0 −12
59B  10 3\sqrt{2}  −4 −12  −4  4  11  −5 −15  9
61A  2 −10  6 −10  6  −2 −12  12  −8  8
61B  2 −10  −2 −10 −10  14  −5 −13  7  −1
71A  −12 6\sqrt{2}  0  0  8  8  5  −3 −13  3
71B  −12 −6\sqrt{2}  −8  12  0  −8  12  −8  −8  12
79A  0  8  −16  8  8  0 −14  10 −10 −10
79B  0  8  −6 −10  4  −16 −14 −10  10  14
89A  −10 3\sqrt{2}  6 −18  2  −6  10 −10 −10  −6
89B  −10 −3\sqrt{2}  −14  6  10  6 −11  15  −5  9

11A · 31B
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})  a_{𝔭}(f_{5})
2  1  1  −1  1 − 2ω_{5}  1
3  −2  −6  −4  0  −2ω_{17}
\sqrt{5}  2  2  −4  2ω_{5}  2
7  −6 −14  −6  2 − 4ω_{5} −2 − 4ω_{17}
11A  −1  −1  1  1  −1
11B  0  4  −6  2 −2 + 2ω_{17}
19A  4  8  0 −4 − 2ω_{5}  2 + 2ω_{17}
19B  0  4  6 −2 + 4ω_{5} −2 + 2ω_{17}
29A  −2  −2  2  2 − 4ω_{5}  −2
29B  −2 −10  6 −2 + 4ω_{5}  2 − 4ω_{17}
31A  −4  −8  −6  2 −2 − 2ω_{17}
31B  −1  −1  −1  1  −1
41A  2  10  −2  2 −2 + 4ω_{17}
41B  2  2  −6  2  2
59A  −4  −4 −12  4 − 8ω_{5}  −4
59B  4  4  8 −8 − 4ω_{5}  4
61A  −2 −10  2  2  2 − 4ω_{17}
61B  10  14 −12  −8  8 + 2ω_{17}
71A  8  4  −4 −8 + 10ω_{5} 10 − 2ω_{17}
71B  4  −4  10 2 − 10ω_{5}  8 − 4ω_{17}
79A  0  0  −4 −12 + 4ω_{5}  0
79B  0  4  −4 −8 + 6ω_{5} −2 + 2ω_{17}
89A  −6 −10 −10  2 + 6ω_{5} −4 − 2ω_{17}
89B  −6  2  −6  2 − 4ω_{5} −10 + 4ω_{17}

349A

𝔭  a_{𝔭}(f_{1})  a_{𝔭}(f_{2})  a_{𝔭}(f_{3})
2  −2 − ω_{5}  1 − \sqrt{2}  ω_{21}
3  −4 − ω_{5}  3  −ω_{21}
\sqrt{5} −3 + 2ω_{5}  2\sqrt{2}  1
7  −2 − 3ω_{5} −4 + 4\sqrt{2}  3ω_{21}
11A  2 − 5ω_{5}  −2  2 − ω_{21}
11B −3 + 2ω_{5}  1 − \sqrt{2}  −3
19A −3 + 6ω_{5} −2 − 2\sqrt{2} 5 − 2ω_{21}
19B −3 − 2ω_{5} −2 − 2\sqrt{2} 5 − 2ω_{21}
29A −6 + 3ω_{5} −4 + 2\sqrt{2}  ω_{21}
29B  1 + ω_{5}  4 + 4\sqrt{2} −5 + ω_{21}
31A  2 − 3ω_{5} −2 + 2\sqrt{2}  2 + ω_{21}
31B  1 + 3ω_{5}  2 − 4\sqrt{2} −3 − ω_{21}
41A  7 − 5ω_{5} −6 + 2\sqrt{2} −3 + 3ω_{21}
41B  2 + ω_{5}  8 + 2\sqrt{2} −8 − ω_{21}
59A  7 − 4ω_{5}  4 −5 + 4ω_{21}
59B  −4ω_{5}  1 + \sqrt{2}  −4ω_{21}
61A −13 + 4ω_{5}  3 − 6\sqrt{2}  7
61B −8 + 3ω_{5}  −8  2 + ω_{21}
71A  −6 −5 + 7\sqrt{2}  2
71B  1 + ω_{5} −8 − 6\sqrt{2} −3 + 5ω_{21}
79A −3 4 + 2\sqrt{2} 5
79B −1 + 4ω_{5} −3 − 5\sqrt{2} −5 + 6ω_{21}
89A  6 − 6ω_{5} −3 + 10\sqrt{2} 10 − 2ω_{21}
89B  11 − ω_{5}  −12\sqrt{2}  5 − ω_{21}

2 · 89B  3 · 41B
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})  a_{𝔭}(f_{5}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{4})  a_{𝔭}(f_{5})
2  −1  −1  −1  −1  1  −2  1  1  −2  1 + ω_{17}
3  −2  0  0  0 −2 − 2\sqrt{3}  −1  −1  1  −1  1
\sqrt{5}  −4  3  2  2 − 2ω_{5}  1 + \sqrt{3}  −3  2  −2  −1  2 − ω_{17}
7  −10  0  0  0 −6 − 2\sqrt{3}  −10  6  2  −6 2 − 4ω_{17}
11A  −2  2  2  2  −2\sqrt{3}  −5  −4  0  5  −ω_{17}
11B  −6  −2  0  4ω_{5} 2 + 2\sqrt{3}  0  4  4  −2  −4
19A  −2  −4  −2 −2 + 4ω_{5}  −2  −8  0  8  6  −4
19B  4  0  0  0  −2  3  −4  4  −1  −ω_{17}
29A  8  −3  −4 −4 − 2ω_{5}  3 + \sqrt{3}  −2  2  −2  2  −2
29B  −6  4  2  2 − 4ω_{5} −4 − 2\sqrt{3}  −1  6 −10  9  2 + ω_{17}
31A  0  −1  −2 −2 − 2ω_{5}  5 + \sqrt{3}  6  −8  −4  4  2ω_{17}
31B  0  −2  0  4ω_{5} −2 − 2\sqrt{3}  6  −4  4 −10  −2ω_{17}
41A  −6  2  2  2  4\sqrt{3}  −6  6  −6  0 −6 + 4ω_{17}
41B  −6  −2  0  4ω_{5} −4 − 4\sqrt{3}  1  −1  1  −1  1
59A  8 −14 −12 −12 + 4ω_{5} 4 − 2\sqrt{3}  −8  4  −8  4 4 + 4ω_{17}
59B  10  −8  −4 −4 + 8ω_{5} −4 + 4\sqrt{3}  7  0  0  −3  −5ω_{17}
61A  −10  10  6  6 − 8ω_{5} −2 + 4\sqrt{3}  6  10 −10  −8 −2 + 4ω_{17}
61B  12  −1  −2 −2 − 2ω_{5} 3 + 3\sqrt{3}  8  −6 −10  6 6 − 2ω_{17}
71A  0  14  8 8 − 12ω_{5} −6 − 2\sqrt{3}  −6  0  −8 −12  −2ω_{17}
71B  −16  −8  −8  −8 4 − 4\sqrt{3}  −9 −12  12  9 −4 + 7ω_{17}
79A  0  15  10 10 − 10ω_{5} −1 + 3\sqrt{3}  −12  8 −12  −4 8 − 4ω_{17}
79B  −4 −11  −8 −8 + 6ω_{5} 1 + 7\sqrt{3}  1  −8  16 −11  4 + ω_{17}
89A  −6 −10 −10  −10  2  −7  2  10  −1  6 − ω_{17}
89B  1  −1  −1  −1  1  6  −2  −6  −8 −6 + 8ω_{17}

379A

𝔭  a_{𝔭}(f_{1})  a_{𝔭}(f_{2})  a_{𝔭}(f_{3})
2  ω_{5}  −2 − ω_{5}  2 − \sqrt{3}
3  1 − 2ω_{5} −3 + 4ω_{5}  2 + \sqrt{3}
\sqrt{5}  1 + ω_{5}  −3 + ω_{5}  −2
7 −3 + 3ω_{5}  −1 − ω_{5}  6 + 3\sqrt{3}
11A  3 − 2ω_{5} −1 − 2ω_{5}  −2
11B −2 − ω_{5}  3ω_{5}  2 + 2\sqrt{3}
19A  1  −7  2 + 2\sqrt{3}
19B  1  1 − 4ω_{5}  1
29A  2 − 2ω_{5} −2 + 6ω_{5}  −3
29B  3 − 6ω_{5} −1 − 4ω_{5} −2 − 2\sqrt{3}
31A −3 + 7ω_{5} −1 + 3ω_{5}  4\sqrt{3}
31B −5ω_{5} −4 + 3ω_{5} 2 − 4\sqrt{3}
41A −6 + 2ω_{5} −2 + 6ω_{5}  2 + 6\sqrt{3}
41B 5 − 4ω_{5} 5 − 4ω_{5} −6 − 2\sqrt{3}
59A −3 + 5ω_{5}  1 − 3ω_{5} −6 + 2\sqrt{3}
59B  3  7 − 6ω_{5}  −8
61A −5 + 2ω_{5}  3 − 6ω_{5}  0
61B −2 + 9ω_{5}  10 + ω_{5} −8 − 2\sqrt{3}
71A −6 + 8ω_{5} −2 + 4ω_{5}  6 − 5\sqrt{3}
71B −5 + 4ω_{5} 7 − 4ω_{5} 1 − 8\sqrt{3}
79A  8 + 3ω_{5} −14 + ω_{5} −10 + 4\sqrt{3}
79B  9 − ω_{5}  3 + 7ω_{5} −6 − 3\sqrt{3}
89A −7 + 8ω_{5}  5 + 2ω_{5}  2
89B  4 + 2ω_{5} −12 + 6ω_{5}  7 − 4\sqrt{3}

2^{2}.\sqrt{5}^{2}  2.101A  3^{2}.\sqrt{5}
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{2}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{4}) a_{𝔭}(f_{1})  a_{𝔭}(f_{2})  a_{𝔭}(f_{3})
2  −  −  −  −  −  −  −3  2  2
3  2  0  −4  0  0  4  −  −  −
\sqrt{5}  −  −  −3  −2  1  2  −  −  −
7  10  0  −6  −4 −10  −4  −14  −4  −4
11A  0  4 − 4ω_{5}  −6  −1  2  −1  4  −3ω_{5}  3 − 3ω_{5}
11B  0  4ω_{5}  −1  6  −3  −2  4 −3 + 3ω_{5}  3ω_{5}
19A  4  0  −1  2  5  −2  4 −4 + 6ω_{5}  2 − 6ω_{5}
19B  4  0  1  4  −5  0  4 2 − 6ω_{5} −4 + 6ω_{5}
29A  −6  0  1  −4  5  0  2 6 − 6ω_{5}  −6ω_{5}
29B  −6  0  0  9  0  3  2  6ω_{5} −6 + 6ω_{5}
31A  −4  4 − 4ω_{5}  0  −5  −8  5  0 −1 − 3ω_{5} −4 + 3ω_{5}
31B  −4  4ω_{5}  9 −11  7  −1  0 −4 + 3ω_{5} −1 − 3ω_{5}
41A  6 −10 + 4ω_{5}  5  3  −3  −1  −10 −6 − 3ω_{5}  9 − 3ω_{5}
41B  6 −6 − 4ω_{5}  0  −2  −8  −6  −10 −9 + 3ω_{5}  6 + 3ω_{5}
59A  −12  0  −8  6  0  14  4  6  −6
59B  −12  0  −14  −5  10  −9  4  6  −6
61A  2 −6 − 4ω_{5}  −7  −6  −3 −10  −2 2 − 3ω_{5} −1 + 3ω_{5}
61B  2 −10 + 4ω_{5}  −12  1  −8  7  −2 −1 + 3ω_{5}  2 − 3ω_{5}
71A  −12  4ω_{5}  16 −12  12  −4  8 3 + 3ω_{5} −6 + 3ω_{5}
71B  −12  4 − 4ω_{5}  −10 −16  2  −8  8 6 − 3ω_{5} −3 − 3ω_{5}
79A  −8  0  10  −8  10  −8  0 8 − 6ω_{5}  2 + 6ω_{5}
79B  −8  0  −12  6  0 −10  0 2 + 6ω_{5}  8 − 6ω_{5}
89A  6  0  −7  9 −15  17  6 6 + 6ω_{5} −12 + 6ω_{5}
89B  6  0  −4  12  0  0  6 12 − 6ω_{5} −6 − 6ω_{5}

11A.41A  2^{2}.29A
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})  a_{𝔭}(f_{5}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})
2  −1  3  −2  1  −ω_{41}  −  −  −  −
3  1  1  2  ω_{41}  ω_{41}  4  −4  0  2ω_{5}
\sqrt{5}  3  −1  −3  −ω_{41}  1  0  −4  4  −2ω_{5}
7  −4  4  8 −2 − 2ω_{41}  0  2  −2  2 −6 + 4ω_{5}
11A  −  −  −  1  1  0  −2  −4  2 + 2ω_{5}
11B  0  0  −5  4 −3 + ω_{41}  0  −4  0  0
19A  2  −6  −5  2ω_{41}  5 − ω_{41}  −4  −4  4  4
19B  5  1  −2 −6 + ω_{41}  −ω_{41}  2  4  −6  4 − 4ω_{5}
29A  −6  −6  7  −2  5 − ω_{41}  −  −  −  −
29B  −6  2  −4  −2  2ω_{41}  6  −6  −2  2 + 4ω_{5}
31A  8  0  −2  0 2 + 2ω_{41}  −10  0  −2  4ω_{5}
31B  −4  4 −10  0 2 + 2ω_{41}  8  4  −8 −4 − 4ω_{5}
41A  −  −  −  1  1  6  2  6  −2
41B  0  0  8 6 − 2ω_{41}  −8  0  0 −12 −4 − 2ω_{5}
59A  −6  2  −3  −2ω_{41} −5 + ω_{41}  −12  −2  0  2 − 6ω_{5}
59B  −9  −1  −4  6 − ω_{41} −10 − ω_{41}  6  −8  6  8ω_{5}
61A  14  6  −4 −2 + 4ω_{41} −8 − 2ω_{41}  −10  10  6 −10 + 4ω_{5}
61B  5  −3  −8  4 + ω_{41}  2 − ω_{41}  8 −12  −4 −8 − 2ω_{5}
71A  −6  −6  1 −4 − 2ω_{41}  7 + ω_{41}  −12  −6  16 −2 + 2ω_{5}
71B  12 −12  6  −4ω_{41} 2 − 2ω_{41}  −6  4  10  −4
79A  −10  14  7 −4 − 2ω_{41}  5 − ω_{41}  −4  −8  −4  4 − 8ω_{5}
79B  −13  11  0  6 + ω_{41} −10 + ω_{41}  8  10  4 −2 + 2ω_{5}
89A  9  −3  0 −4 − ω_{41} −10 + ω_{41}  −6 −10  −6  2 + 4ω_{5}
89B  0 −16  13 6 + 2ω_{41} −5 − ω_{21}  −18  −2  6 14 − 4ω_{5}

\sqrt{5}^{2}.19A  2.11A.11B
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})  a_{𝔭}(f_{5}) a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3}) a_{𝔭}(f_{4}) a_{𝔭}(f_{5})
2  −3  1  3 −1 − \sqrt{6}  1 − \sqrt{6}  −  −  −  −  −
3  −2  2  2  1 + \sqrt{6} −1 + \sqrt{6}  2  1  1  5  −3
\sqrt{5}  −  −  −  −  −  2  0  0  −4  4
7  4  −2  −4 8 − 2\sqrt{6} −8 − 2\sqrt{6}  14  −1  −1  5  13
11A  −2  0  −2  −2\sqrt{6}  2\sqrt{6}  −  −  −  −  −
11B  −6  0  −6  3 + \sqrt{7}  3 − \sqrt{6}  −  −  −  −  −
19A  −  −  −  −  −  −4  −1  5  5  7
19B  −4  4  4 2 + 2\sqrt{6} −2 + 2\sqrt{6}  −4  5  −1  5  7
29A  −4  −6  4 −3 + 2\sqrt{6} 3 + 2\sqrt{6}  10  3  −3  −5  −3
29B  10  6 −10  0  0  10  −3  3  −5  −3
31A  2  −4  2  2  2  −8  −7  5  7  −5
31B  −4  8  −4 −1 + 2\sqrt{6} −1 − 2\sqrt{6}  −8  5  −7  7  −5
41A  −2  −6  −2  −2\sqrt{6}  2\sqrt{6}  −2  0  0  −8  −8
41B  −2  −6  −2  −2\sqrt{6}  2\sqrt{6}  −2  0  0  −8  −8
59A  4 −12  −4 3 − 2\sqrt{6} −3 − 2\sqrt{6}  12  0 −12  0 −12
59B  −12 −12  12 −9 + \sqrt{6}  9 + \sqrt{6}  12 −12  0  0 −12
61A  −2 −10  −2 5 + 3\sqrt{6} 5 − 3\sqrt{6}  −10  −1  −7 −13  5
61B  −6  14  −6 −7 + \sqrt{6} −7 − \sqrt{6}  −10  −7  −1 −13  5
71A  12  0  12  −3  −3  0  9  −3  −3  −7
71B  −10  12 −10 −9 − \sqrt{6} −9 + \sqrt{6}  0  −3  9  −3  −7
79A  −8  16  8 −1 − \sqrt{6}  1 − \sqrt{6}  −12  8  −4  0  4
79B  0  −8  0  5  −5  −12  −4  8  0  4
89A  8  6  −8 6 − 4\sqrt{6} −6 − 4\sqrt{6}  −18  9 −15  15  7
89B  2  −6  −2 −6 + 4\sqrt{6} 6 + 4\sqrt{6}  −18 −15  9  15  7

499A

𝔭  a_{𝔭}(f_{1})  a_{𝔭}(f_{2})  a_{𝔭}(f_{3})  a_{𝔭}(f_{4})
2  −1 + 2ω_{5}  −2  3  0
3  3 − ω_{5} 2 − 4ω_{5} −3 + 3ω_{5}  0
\sqrt{5}  2 − 2ω_{5} −3 + ω_{5} 2 − 2ω_{5}  −ω_{29}
7  6 + 3ω_{5} 2 − 8ω_{5} 4 − 11ω_{5} 2 + 2ω_{29}
11A −2 − 2ω_{5} −2 − ω_{5} −2 + 2ω_{5}  3 + ω_{29}
11B 4ω_{5} −2 + 2ω_{5} −4 2
19A  2 + ω_{5} −6 + 2ω_{5} 4 − 5ω_{5} 2 + 2ω_{29}
19B  4 − 3ω_{5} −4 + 3ω_{5} −4 + 5ω_{5}  1 + ω_{29}
29A  −7 − ω_{5} −3 + 3ω_{5}  1 − ω_{5}  6 + ω_{29}
29B −4 + 3ω_{5} −3 − ω_{5} −4 + 3ω_{5} 2 − 3ω_{29}
31A 7 − 5ω_{5} −4 + 7ω_{5} 3 − ω_{5} −7 + ω_{29}
31B −2 − 2ω_{5} 2 − 3ω_{5} 6 − 2ω_{5}  1 − ω_{29}
41A −2 + 3ω_{5}  0 −4 + ω_{5} −6 + 2ω_{29}
41B  4ω_{5} 5 − 7ω_{5} −8 + 4ω_{5} −4 − ω_{29}
59A  −2 − ω_{5} −4 + 8ω_{5} −4 + ω_{5} 6 − 4ω_{29}
59B  3 − ω_{5} 8 − 4ω_{5} 3 − 9ω_{5}  0
61A  3 + 3ω_{5}  0 1 + 3ω_{5}  −8
61B −4 + 2ω_{5} −2 + 3ω_{5} −4 + 6ω_{5} −1 − 3ω_{29}
71A −14 + 2ω_{5} 7 + ω_{5} 2 − 2ω_{5} 6 − ω_{29}
71B −4 + 2ω_{5} 2 + 2ω_{5} 4 − 6ω_{5}  −8
79A −4 + 3ω_{5} −8 + 5ω_{5} 4 + 7ω_{5} −1 − ω_{29}
79B  11 − 7ω_{5} −6 + 4ω_{5} 5 + 5ω_{5} 4 + 4ω_{29}
89A  −7 − ω_{5} −5 + ω_{5} 5 − 5ω_{5} −6 − ω_{29}
89B −2 − 6ω_{5} 4 − 8ω_{5} −6 + 2ω_{5} 8 − 2ω_{29}

3.\sqrt{5}.11A  2.\sqrt{5}^{3}
𝔭 a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})  a_{𝔭}(f_{4})  a_{𝔭}(f_{1})  a_{𝔭}(f_{2})
2  3  −3  1 −1 + 2ω  −  −
3  −  −  −  −  0  0
\sqrt{5}  −  −  −  −  −  −
7  −2  2  2  2 − 4ω_{5}  10  −10
11A  −  −  −  −  2  2
11B  0  −4  4  4 − 4ω_{5}  2  2
19A  0  −8  −4  0  −5ω_{5}  5 − 5ω_{5}
19B  4  −4  −4  0  −5 + 5ω_{5}  5ω_{5}
29A  2  6  −2  2 − 4ω_{5}  −5ω_{5}  5 − 5ω_{5}
29B  −2  −6  −2  2 − 4ω_{5}  −5 + 5ω_{5}  5ω_{5}
31A  4  −8  0  4 − 4ω_{5}  −5ω_{5} −3 + 5ω_{5}
31B  −8  4  0  4ω_{5}  −3 + 5ω_{5}  2 − 5ω_{5}
41A  2  6  10 −10 + 4ω_{5}  2 − 5ω_{5} −3 + 5ω_{5}
41B  −10 −10  −6  2  −3 + 5ω_{5}  2 − 5ω_{5}
59A  0  4  12 −12 + 4ω_{5}  5 + 5ω_{5} −10 + 5ω_{5}
59B  4  12  −4  4 − 8ω_{5}  10 − 5ω_{5} −5 − 5ω_{5}
61A  −6  −6  −2  2  2 − 10ω_{5} −8 + 10ω_{5}
61B  −10  −2  −2 −2 + 8ω_{5} −8 + 10ω_{5} 2 − 10ω_{5}
71A  0  8  16  −8  −8 + 5ω_{5} −3 − 5ω_{5}
71B  −8  −4  0 −4 + 12ω_{5}  −3 − 5ω_{5} −8 + 5ω_{5}
79A  −16  0  −8 −4 + 8ω_{5}  −10ω_{5} 10 − 10ω_{5}
79B  4 −12  −8  4 − 8ω_{5} −10 + 10ω_{5}  10ω_{5}
89B  −10  2  −6 −2 + 4ω_{5}  −10  10
89A  14  −6  −6 −2 + 4ω_{5}  −10  10

13 : x^{3} + 6x^{2} 151A : x^{3} − x^{2} 181A : x^{4} − 12x^{2}
+8x − 2  −5x + 1  +8x − 1
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{1})  a_{𝔭}(f_{1})
2 (3, −2, −1)  (0, 1, 0)  (0, 1, 0, 0)
3 (2, 10, 3)  (6, 1, −3)  (−3, 0, 1, 0)
\sqrt{5} (−4, −2, 0)  (−2, −1, 1)  (−2, 12, −1, −2)
7 (6, −2, 0)  (4, −1, 1)  (6, −38, 2, 6)
11A (0, 1, 0)  (−4, −2, 2)  (8, −20, 0, 3)
11B (0, 1, 0)  (5, −1, −2)  (8, −33, 2, 5)
19A (−2, −1, 0)  (−6, 0, 2)  (6, −31, 2, 5)
19B (−2, −1, 0)  (3, 1, −2)  (−6, 26, −2, −4)
29A (−6, −12, −3) (−12, −2, 4)  (−12, 50, −4, −8)
29B (−6, −12, −3) (−6, 2, 2)  (−6, 2, 2, 0)
31A (−4, −7, −2) (−4, 2, 2)  (2, 6, −2, −1)
31B (−4, −7, −2) (8, 5, −3)  (−10, 50, −4, −8)
41A (4, −4, −2)  (−8, −3, 5)  (−7, 28, −1, −4)
41B (4, −4, −2)  (8, 2, −4)  (−10, 26, 0, −4)
59A (10, −3, −2) (−16, −3, 7)  (−6, −5, 2, 1)
59B (10, −3, −2) (4, −4, 0)  (22, −60, 2, 9)
61A (2, −10, −3) (8, 6, −4)  (18, −12, −2, 2)
61B (2, −10, −3) (8, 1, −5)  (−8, 40, −1, −6)
71A (−10, 17, 6)  (18, 1, −5)  (12, −14, 0, 2)
71B (−10, 17, 6)  (2, 4, −2)  (2, −12, 0, 2)
79A (12, 10, 2)  (−2, −4, 2)  (22, −56, 2, 9)
79B (12, 10, 2)  (14, 1, −7)  (16, −34, 0, 5)
89A (2, 16, 4)  (6, 2, −6)  (−12, 10, 0, −2)
89B (2, 16, 4)  (−17, −7, 8)  (−4, −2, 1, 0)

For ℚ(α) ≅ ℚ[x]/(x^{3} − x^{2} − 5x + 1), the chosen integral basis is

{1, α, 1/2(α^{2} + 1)}.

For ℚ(α) ≅ ℚ[x]/(x^{4} − 12x^{2} + 8x − 1) the chosen integral basis is

211A : x^{4} − 2x^{3}  229A : x^{3} − 3x^{2}
−8x^{2} + 6x + 7  −x + 1
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})
2 (0, 1, 0, 0)  −3 (0, 1, 0)
3 (−3, 5, 4, −3)  −1 (2, 2, −1)
\sqrt{5} (3, −2, −2, 1)  −4 (1, −4, 1)
7 (−1, −8, −2, 3)  −10 (0, −10, 4)
11A (1, −3, −1, 1)  −2 (1, 8, −3)
11B (−4, −2, 2, 0)  0 (−3, −2, 1)
19A (4, 3, −1, −1)  −5 (−3, 5, −1)
19B (4, −2, −4, 2)  −7 (2, −1, 0)
29A (−2, −4, −2, 2)  6 (−1, 2, −1)
29B (−12, 3, 7, −3)  3 (2, −4, 1)
31A (−7, 5, 6, −3)  −3 (3, 7, −3)
31B (6, 5, −1, −1)  8 (−3, −8, 3)
41A (−3, 1, 2, −1)  2 (1, −6, 1)
41B (14, −10, −12, 6)  −9 (−1, 4, 0)
59A (−9, 3, 5, −3)  9 (−8, −1, 2)
59B (−2, 12, 4, −4)  −10 (−1, −12, 3)
61A (0, 12, 4, −4)  0 (4, 14, −6)
61B (4, −4, −6, 4)  −14 (−8, −6, 4)
71A (12, −14, −8, 6)  −3 (−6, −19, 6)
71B (5, 0, 0, −1)  16 (−9, 2, 1)
79A (−10, −8, −2, 4)  6 (7, 22, −7)
79B (−9, 7, 5, −3)  −12 (9, −10, 3)
89A (0, 2, −2, 0)  −5 (5, 14, −4)
89B (−5, 3, 7, −3)  −6 (7, 4, −3)

For ℚ(α) ≅ ℚ[x]/(x^{4} − 2x^{3} − 8x^{2} + 6x + 7), the chosen integral basis is

{1, α, 1/2(α^{2} + 1), 1/4(α^{3} + α^{2} + α + 1)}.

241A : x^{5} − 3x^{4} − 7x^{3}  251A : x^{3} + x^{2}
+20x^{2} − 4x − 8  −5x + 1
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})
2 (0, 1, 0, 0, 0)  −3 (0, 1, 0)
3 (−8, −4, 7, 3, −3)  0 (3, −2, −1)
\sqrt{5} (0, −7, 3, 3, −2)  −3 (−3, 1, 1)
7 (−6, 8, 2, −2, 0)  0 (9, −2, −3)
11A (14, 28, −18, −14, 10)  −6 (5, −2, −1)
11B (4, −4, −3, 0, 1)  −4 (5, −2, −1)
19A (12, 30, −18, −14, 10)  0 (−5, 0, 1)
19B (−19, −19, 17, 10, −8)  8 (3, 0, −1)
29A (10, 16, −12, −8, 6)  −5 (−9, 2, 2)
29B (−4, −26, 12, 12, −8)  8 (−5, 2, 1)
31A (−8, −16, 13, 9, −7)  1 (−7, 2, 2)
31B (12, 16, −12, −8, 6)  −8 (8, −2, −2)
41A (−21, −47, 29, 22, −16)  −10 (8, 0, −2)
41B (−12, −10, 11, 6, −5)  0 (−11, 4, 3)
59A (−14, −28, 14, 12, −8)  −11 (−1, 3, 1)
59B (−26, −54, 32, 26, −18)  −9 (−6, 4, 1)
61A (−4, 10, 2, −2, 0)  −7 (−4, −1, 2)
61B (24, 48, −30, −22, 16)  8 (0, 4, 0)
71A (32, 46, −35, −25, 19)  4 (−11, 10, 3)
71B (−4, 12, 2, −2, 0)  3 (−4, 6, 1)
79A (−24, −29, 25, 15, −12)  −2 (−12, 0, 2)
79B (−16, −18, 17, 12, −9)  1 (−14, 3, 4)
89A (11, 13, −7, −6, 4)  8 (13, −10, −5)
89B (34, 24, −30, −16, 14)  4 (17, −4, −3)

For ℚ(α) ≅ ℚ[x]/(x^{5} − 3x^{4} − 7x^{3} + 20x^{2} − 4x − 8), the chosen integral basis is

{1, α, α^{2}, 1/2(α^{3} + α^{2} + α), 1/4(α^{4} + α^{3} + α^{2})}.

3 · 29B : x^{3} + x^{2} − 5x + 2 271A : x^{4} − 14x^{2} + 8x + 25
𝔭  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})
2  0 (3, −3, −1)  1 (0, 1, 0, 0)
3  −1 (1, 0, 0)  −2 (−5, −5, 1, 2)
\sqrt{5}  2 − \sqrt{21} (0, 1, 0)  2 (1, 2, 0, −1)
7  −9 − \sqrt{21} (2, 2, 1)  2 (−6, 8, 2, −2)
11A  2\sqrt{21} (0, 2, 0)  4 (−2, −6, 0, 2)
11B  1 + \sqrt{21} (−8, 4, 3)  −4 (11, 4, −2, −2)
19A −2 + 2\sqrt{21} (4, −4, −2)  −8 (6, 3, −1, −1)
19B −4 − \sqrt{21} (2, 3, 0)  −4 (−8, −5, 2, 2)
29A  1 − \sqrt{21} (−6, 0, 1)  −2 (5, 0, −2, 0)
29B  −1 (1, 0, 0)  −4 (6, 0, 0, 0)
31A  5 − 3\sqrt{21} (−4, 0, 1)  8 (−4, −5, 0, 2
31B −2 + 4\sqrt{21} (4, −2, −2)  8 (2, 10, 0, −4)
41A  5 − 3\sqrt{21} (−6, 2, 1)  6 (10, 7, −3, −3)
41B  2 (10, −8, −4)  −10 (−2, −7, 1, 3)
59A −3 + 3\sqrt{21} (−4, −2, −1)  8 (−9, 1, 1, 0)
59B 8 − 3\sqrt{21} (−14, 7, 4) −4 (−1, 4, 0, −2)
61A −6 − 2\sqrt{21} (18, −6, −4)  −14 (9, 5, −1, −2)
61B  2 − 5\sqrt{21} (12, −5, −4)  10 (23, 6, −6, −3)
71A  6 + \sqrt{21} (18, −11, −6)  −8 (10, 6, −2, −2)
71B −4 − 4\sqrt{21} (−8, 8, 4)  0 (−14, −2, 4, 2)
79A 8 + 2\sqrt{21} (−12, 10, 4) 16 (−8, −12, 2, 4)
79B −6 + \sqrt{21} (−10, 3, 2)  12 (−5, 6, 2, −1)
89A 4 + \sqrt{21} (−12, 5, 2) 2 (−20, −6, 4, 4)
89B −6 − 4\sqrt{21} (26, −16, −8)  −14 (2, 6, 0, −2)

For ℚ(α) ≅ ℚ[x]/(x^{4} − 14x^{2} + 8x + 25), the chosen integral basis is

{1, α, 1/2(α^{2} + 1), 1/4(α^{3} + α^{2} + α + 1)}.

281A : x^{5} − x^{4} − 11x^{3}+ 17 : x^{4} − 4x^{3} − 2x^{2} + 8x + 1
8x^{2} + 28x − 8
𝔭 a_{𝔭}(f_{1})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})
2 (0, 1, 0, 0, 0)  −3 (0, 1, 0, 0)
3 (2, 2, −2, −2, 2)  −6 (−1, −3, 2, 0)
\sqrt{5} (2, 2, 1, 0, −1)  −2 (6, 0, −5, 1)
7 (7, 7, −1, −2, 0)  2 (−23, 7, 24, −6)
11A (2, −12, −4, 2, 2)  0 (−4, 0, 4, −1)
11B (−4, −2, 2, 1, −1)  0 (−4, 0, 4, −1)
19A (2, −8, −2, 2, 0)  −4 (−6, 4, 6, −2)
19B (−10, 11, 8, −1, −4)  −4 (−6, 4, 6, −2)
29A (3, −1, −5, −2, 4)  6 (8, −6, −6, 2)
29B (−4, 4, 4, 1, −3)  6 (8, −6, −6, 2)
31A (5, −1, −5, −2, 4)  4 (−4, 4, −1, 0)
31B (6, −5, −2, 1, 0)  4 (−4, 4, −1, 0)
41A (6, 10, 0, −2, 0)  −6 (10, −4, −11, 3)
41B (−6, 15, 6, −3, −2)  −6 (10, −4, −11, 3)
59A (−8, 6, 4, 0, −2)  −12 (2, 0, 0, 0)
59B (−8, 8, 8, 2, −6)  −12 (2, 0, 0, 0)
61A (−4, 20, 6, −4, −2)  −10 (8, −6, −3, 1)
61B (−12, 2, 6, 1, −3)  −10 (8, −6, −3, 1)
71A (10, 6, 0, 0, −2)  −4 (−6, 6, 2, −1)
71B (2, 8, 3, −2, −1)  −4 (−6, 6, 2, −1)
79A (6, −22, −10, 4, 4)  12 (22, −2, −17, 4)
79B (−8, −18, −4, 4, 2)  12 (22, −2, −17, 4)
89A (4, −2, 2, 2, −2)  10 (1, 5, −4, 0)
89B (−2, −16, −3, 4, 1)  10 (1, 5, −4, 0)

For ℚ(α) ≅ ℚ[x]/(x^{5} − x^{4} − 11x^{3} + 8x^{2} + 28x − 8), the chosen integral basis is

{1, α, α^{2}, 1/2(α^{3} + α^{2} + α), 1/4(α^{4} + α^{3} − α^{2} + 2α)}.

For ℚ(α) ≅ ℚ[x]/(x^{4} − 4x^{3} − 2x^{2} + 8x + 1), the chosen integral basis is

\sqrt{5} · 59B : x^{3} + x^{2} − 7x + 1 311A : x^{4} + x^{3} − 7x^{2} − x + 2
𝔭  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})
2  2 − \sqrt{3} (0, 1, 0)  −2 (0, 1, 0, 0)
3  1 + 2\sqrt{3} (−3, 0, 1)  1 (−1, −8, 1, 2)
\sqrt{5}  −1 (1, 0, 0)  −4 (3, 3, −1, −1)
7 −1 − 6\sqrt{3} (−9, 4, 3)  5 (−1, 20, −3, −6)
11A −3 + 2\sqrt{3} (−1, 0, 1)  −5 (1, −8, 1, 2)
11B  −\sqrt{3} (8, −3, −3)  −2 (−2, 8, 0, −2)
19A  2 + 3\sqrt{3} (6, −1, −3)  −2 (−3, −3, 1, 1)
19B  2 (−6, 0, 2)  1 (−1, −1, 0, 0)
29A  −3\sqrt{3} (0, −3, −1)  4 (−2, −14, 2, 4)
29B  \sqrt{3} (0, −3, −1)  −5 (−5, 7, 0, −2)
31A −4 + 2\sqrt{3} (−4, 2, 2)  −4 (4, −10, 0, 2)
31B −7 − 2\sqrt{3} (−1, 4, 1)  −3 (5, 0, −1, 0)
41A −6 − 3\sqrt{3} (2, −1, −1)  8 (3, 5, −1, −1)
41B  4\sqrt{3} (−4, 4, 2)  −10 (−4, −6, 2, 2)
59A  −4\sqrt{3} (−12, 4, 4)  1 (−1, 12, −1, −2)
59B  −1 (1, 0, 0)  14 (10, 4, −4, −2)
61A  2 − 3\sqrt{3} (2, 3, −1)  4 (3, 19, −3, −5)
61B −4 + 6\sqrt{3} (8, −2, −4)  7 (−5, −2, 1, 0)
71A  −2\sqrt{3} (−16, 2, 6)  −10 (11, 9, −3, −3)
71B  12 − \sqrt{3} (−4, 1, 1)  −13 (1, 19, −2, −4)
79A  5 − 4\sqrt{3} (13, −2, −3)  −14 (6, 14, −4, −4)
79B  14 (−18, 4, 6)  −8 (−9, 5, 1, −1)
89A  −2\sqrt{3} (16, 2, −4)  0 (4, 22, −2, −6)
89B  −12 (0, 0, −2)  0 (−16, −4, 4, 2)

For ℚ(α) ≅ ℚ[x]/(x^{3} + x^{2} − 7x + 1), the chosen integral basis is

{1, α, 1/2(α^{2} + 1)}.

For ℚ(α) ≅ ℚ[x]/(x^{4} + x^{3} − 7x^{2} − x + 2), the chosen integral basis is

{1, α, α^{2}, 1/2(α^{3} + α)}.

\begin{aligned}& 331A : x^{6} − 2x^{5} − 17x^{4} + 32x^{3} \\ & +39x^{2} − 86x + 25 \\ 𝔭 & a_{𝔭}(f_{1}) \\ 2 & (0, 1, 0, 0, 0, 0) \\ 3 & (15, −19, −8, 12, 4, −11) \\ \sqrt{5} & (−1, 3, 1, −3, −1, 3) \\ 7 & (−30, 43, 13, −23, −7, 20) \\ 11A & (−4, 13, 1, −8, −2, 7) \\ 11B & (−16, 13, 11, −7, −3, 6) \\ 19A & (21, −21, −12, 14, 5, −13) \\ 19B & (14, −26, −6, 14, 4, −12) \\ 29A & (−32, 34, 17, −20, −7, 18) \\ 29B & (6, −6, −6, 4, 2, −4) \\ 31A & (10, 2, −6, −2, 0, 2) \\ 31B & (−30, 34, 17, −20, −7, 18) \\ 41A & (2, 22, −6, −12, −2, 10) \\ 41B & (7, −13, −4, 6, 2, −5) \\ 59A & (20, −30, −12, 18, 6, −16) \\ 59B & (−4, 0, −2, −2, 0, 2) \\ 61A & (22, −30, −12, 18, 6, −16) \\ 61B & (6, −11, −1, 5, 1, −4) \\ 71A & (63, −79, −32, 46, 15, −41) \\ 71B & (−44, 36, 26, −20, −8, 18]) \\ 79A & (−51, 57, 27, −33, −11, 29) \\ 79B & (10, 4, −10, −4, 0, 4) \\ 89A & (9, −41, 0, 22, 5, −19) \\ 89B & (−34, 17, 19, −8, −4, 7) \end{aligned} \begin{matrix}11A · 31A : x^{3} − x^{2} − 12x + 16 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 0 & 0 & (0, 1, 0) \\ 0 & 0 & (8, −2, −4) \\ 2 & 1 & (−1, 0, 1) \\ 2 & 0 & (0, 0, −2) \\ 1 & 1 & (−1, 0, 0) \\ 0 & 2 & (−2, 0, 2) \\ −2 & 0 & (4, −2, −2) \\ −1 & 0 & (0, −1, −1) \\ −1 & 0 & (0, 1, −1) \\ −4 & −5 & (7, −2, −1) \\ 1 & 1 & (−1, 0, 0) \\ 1 & 2 & (−18, 5, 7) \\ −2 & −3 & (9, 0, −1) \\ 4 & 2 & (14, −2, −6) \\ −11 & −10 & (−6, 3, 5) \\ 5 & 0 & (−20, 3, 7) \\ 5 & 2 & (−10, −1, 3) \\ 5 & 2 & (14, −3, −7) \\ −2 & 2 & (−22, 6, 8) \\ −2 & 2 & (−2, 2, 0) \\ 6 & 5 & (−21, 6, 9) \\ −4 & −5 & (−1, 0, −1) \\ −4 & 0 & (12, −2, −8) \\ −6 & −10 & (−6, 4, 4) \end{matrix}

For ℚ(α) ≅ ℚ[x]/(x^{6} −2x^{5} −17x^{4} +32x^{3} +39x^{2} −86x+25), the chosen integral

basis is

{1, α, 1/2(α^{2} + 1), 1/2(α^{3} + α), 1/4(α^{4} − 1), 1/28(α^{5} + 2α^{4} − 2α^{3} − 4α^{2} + 9α + 6)}.

For ℚ(α) ≅ ℚ[x]/(x^{3} − x^{2} − 12x + 16), the chosen integral basis is

{1, α, 1/4(α^{2} − α)}.

\sqrt{5} · 71B : x^{3} + x^{2} − 11x + 1  359A : x^{3} − x^{2} − 5x + 1
𝔭  a_{𝔭}(f_{1}) a_{𝔭}(f_{2})  a_{𝔭}(f_{1}) a_{𝔭}(f_{2}) a_{𝔭}(f_{3})
2  1 − \sqrt{2} (0, 1, 0)  2  −2 (0, 1, 0)
3  1 + 3\sqrt{2} (−3, 0, 1)  1  −3 (0, −2, 0)
\sqrt{5}  1 (1, 0, 0)  −3  −3 (2, 0, 0)
7 −2 − 6\sqrt{2} (0, −2, 0)  −4  8 (7, 2, −3)
11A −3 − 2\sqrt{2} (6, −1, −1)  0  2 (0, 0, 0)
11B −1 + 2\sqrt{2} (6, −1, −1)  6  −4 (1, 0, −1)
19A  1 − 3\sqrt{2} (−5, 0, 1)  −4  −4 (9, 1, −4)
19B  6 (8, −2, −2)  2  −2 (−9, −2, 5)
29A −1 + \sqrt{2} (−7, 0, 1)  3  −1 (7, 3, −2)
29B −4 + 2\sqrt{2} (8, 0, −2)  6  4 (0, −2, 0)
31A  6\sqrt{2} (−2, 2, 0)  −4  0 (11, 0, −5)
31B −2 + 2\sqrt{2} (−6, 0, 2)  −7  1 (−11, −3, 4)
41A −8 − 2\sqrt{2} (−8, 0, 2)  −6  0 (−11, 0, 3)
41B −3 − 6\sqrt{2} (0, −1, −1)  −6  −4 (−11, 0, 5)
59A  3 + 5\sqrt{2} (−1, −2, −1)  6  −6 (−10, −2, 4)
59B  7 − \sqrt{2} (−1, 0, 1)  9 −15 (0, 2, 2)
61A −2 − 2\sqrt{2} (4, 2, 0)  11  −5 (−4, 0, −2)
61B  −1 (−2, 3, 1)  −10  12 (13, 6, −7)
71A −6 − 2\sqrt{2} (−10, 4, 2)  0  6 (18, 6, −8)
71B  1 (−1, 0, 0)  −3  9 (−8, 0, 4)
79A  −4 (−16, 4, 4)  −10 −12 (13, 6, −3)
79B  10 (−16, 2, 2)  −4  2 (−14, 0, 6)
89A  2 − 8\sqrt{2} (−6, 0, 0)  −3 −15 (5, 1, −2)
89B −6 + 4\sqrt{2} (18, −4, −4)  −12 −14 (8, 0, −2)

For ℚ(α) ≅ ℚ[x]/(x^{3} + x^{2} − 11x + 1), the chosen integral basis is

{1, α, 1/2(α^{2} + 1)}.

For ℚ(α) ≅ ℚ[x]/(x^{3} − x^{2} − 5x + 1), the chosen integral basis is

{1, α, 1/2(α^{2} + 1)}.

\begin{aligned}& 19A^{2} : x^{4} − 10x^{2} + 5 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 2 & −2 & 2 & (0, 1, 0, 0) \\ 3 & −1 & 1 & (0, −8, 0, 3) \\ \sqrt{5} & −3 & −3 & (3, 0, −1, 0) \\ 7 & 2 & 2 & (−4, 0, 1, 0) \\ 11A & −3 & 3 & (0, 4, 0, −2) \\ 11B & −3 & 3 & (0, 4, 0, −2) \\ 19A & −1 & −1 & (−6, 0, 4, 0) \\ 19B & 0 & 0 & (0, 0, 0, 0) \\ 29A & 6 & −6 & (0, 3, 0, −1) \\ 29B & 3 & 3 & (2, 0, −3, 0) \\ 31A & −7 & −7 & (0, 0, 2, 0) \\ 31B & −5 & 5 & (0, −12, 0, 4) \\ 41A & −3 & −3 & (8, 0, −5, 0) \\ 41B & 3 & −3 & (0, 7, 0, −3) \\ 59A & 12 & −12 & (0, 6, 0, −2) \\ 59B & −3 & −3 & (−18, 0, 12, 0) \\ 61A & −14 & 14 & (0, −16, 0, 7) \\ 61B & −5 & 5 & (0, 11, 0, −3) \\ 71A & 9 & −9 & (0, 10, 0, −4) \\ 71B & 0 & 0 & (0, 10, 0, −4) \\ 79A & 5 & 5 & (10, 0, −8, 0) \\ 79B & 4 & −4 & (−2, 0, −2, 0) \\ 89A & 9 & −9 & (0, 10, 0, −2) \\ 89B & −18 & −18 & (0, −18, 0, 7) \end{aligned} \begin{aligned}& 19 : x^{3} − 2x^{2} − 5x + 4 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) & a_{𝔭}(f_{4}) \\ 3 & −4 & 0 & (0, 1, 0) \\ 6 & −2 & 0 & (2, −2, 0) \\ −2 & 3 & 1 & (3, 2, −1) \\ 10 & −13 & −5 & (11, 2, −1) \\ −4 & 3 & 2 − ω_{15} & (−3, −2, 1) \\ −4 & 3 & 2 + ω_{15} & (−3, −2, 1) \\ −1 & 1 & −1 & (1, 0, 0) \\ −1 & 1 & −1 & (1, 0, 0) \\ 6 & 6 & 2ω_{15} & (−6, 0, 0) \\ 6 & 6 & −2ω_{15} & (−6, 0, 0) \\ −4 & −4 & 2 + 2ω_{15} & (−8, −2, 2) \\ −4 & −4 & 2 − 2ω_{15} & (−8, −2, 2) \\ −10 & −6 & 2 & (10, 2, −2) \\ −10 & −6 & 2 & (10, 2, −2) \\ 0 & −6 & −2ω_{15} & (−2, −2, 0) \\ 0 & −6 & 2ω_{15} & (−2, −2, 0) \\ 2 & −1 & 2 + ω_{15} & (9, 4, −3) \\ 2 & −1 & 2 − ω_{15} & (9, 4, −3) \\ 4 & 6 & −8 − 2ω_{15} & (10, −2, 0) \\ 4 & 6 & −8 + 2ω_{15} & (10, −2, 0) \\ −4 & 8 & −10 − 2ω_{15} & (12, 2, −2) \\ −4 & 8 & −10 + 2ω_{15} & (12, 2, −2) \\ 2 & 12 & −10 + 2ω_{15} & (−16, 4, 2) \\ 2 & 12 & −10 − 2ω_{15} & (−16, 4, 2) \end{aligned}

For ℚ(α) ≅ ℚ[x]/(x^{4} − 10x^{2} + 5), the chosen integral basis is

\begin{aligned}& 389A : x^{4} − 3x^{3} − 3x^{2} + 10x − 4 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) \\ 2 & −2 − ω_{5} & (0, 1, 0, 0) \\ 3 & −3ω_{5} & (2, −1, 0, 0) \\ \sqrt{5} & −3 + 2ω_{5} & (−4, 5, 3, −2) \\ 7 & 1 + 2ω_{5} & (−22, 20, 12, −7) \\ 11A & −3 & (6, −7, −3, 2) \\ 11B & −3 + ω_{5} & (4, −4, −3, 2) \\ 19A & −3 + 3ω_{5} & (10, −4, −5, 2) \\ 19B & −3ω_{5} & (2, −7, −2, 2) \\ 29A & −4 + 7ω_{5} & (16, −19, −8, 6) \\ 29B & −5 − 2ω_{5} & (−5, 1, 3, −1) \\ 31A & −2 + 3ω_{5} & (0, 0, 1, −1) \\ 31B & 3 − 6ω_{5} & (0, 1, −1, 0) \\ 41A & 9 − ω_{5} & (−6, 2, 1, 0) \\ 41B & −8 + 4ω_{5} & (−14, 20, 8, −6) \\ 59A & −3 + 3ω_{5} & (6, 1, −4, 1) \\ 59B & 3 − 2ω_{5} & (−17, 23, 9, −7) \\ 61A & 10 − 3ω_{5} & (1, 3, 0, −1) \\ 61B & 2 − 9ω_{5} & (−16, 9, 8, −4) \\ 71A & 12 − 4ω_{5} & (−11, 16, 6, −5) \\ 71B & −1 + 2ω_{5} & (4, −5, −3, 2) \\ 79A & 2 − 3ω_{5} & (12, −3, −6, 2) \\ 79B & −9 + 8ω_{5} & (−2, 1, 1, 0) \\ 89A & −5 − 8ω_{5} & (4, 6, 4, −3) \\ 89B & −15 + 3ω_{5} & (18, −10, −7, 4) \end{aligned} \begin{matrix}\sqrt{5} · 79A : x^{3} + x^{2} − 9x − 1 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 1 & −3 & (−1, 2, 0) \\ 4 & −2 & (2, −2, −2) \\ −1 & −1 & (1, 0, 0) \\ 4 & −6 & (2, −6, −2) \\ −2 & −4 & (4, −2, −2) \\ 2 & −2 & (0, 0, 2) \\ 0 & 2 & (0, −2, 0) \\ 0 & −8 & (−4, 0, 4) \\ 6 & −2 & (−6, 4, 4) \\ −4 & 0 & (2, −4, −2) \\ 0 & 4 & (4, 0, −4) \\ −4 & 0 & (4, 0, −4) \\ −6 & 12 & (−2, 6, 0) \\ 2 & −10 & (2, −4, 0) \\ 12 & 0 & (−4, −4, 4) \\ 4 & −8 & (−4, 4, 0) \\ −14 & −10 & (−2, 4, 0) \\ −10 & 12 & (2, −6, 0) \\ −6 & 0 & (−8, 6, 6) \\ 4 & 0 & (−4, 8, 0) \\ −1 & 1 & (1, 0, 0) \\ −16 & −8 & (0, 0, −4) \\ 0 & 12 & (−10, 4, 6) \\ 0 & 12 & (−10, 4, 6) \end{matrix}

For ℚ(α) ≅ ℚ[x]/(x^{4} − 3x^{3} − 3x^{2} + 10x − 4), the chosen integral basis is

{1, α, α^{2}, 1/2(α^{3} + α^{2} + α)}.

For ℚ(α) ≅ ℚ[]/(x^{3} + x^{2} − 9x − 1), the chosen integral basis is

{1, 1/2(α + 1)1/4(α^{2} − 1)}.

\begin{aligned}& 401A : x^{6} − 15x^{4}+ \\ & 12x^{3} + 39x^{2} − 24x − 29 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) \\ 2 & −1 & (0, 1, 0, 0, 0, 0) \\ 3 & −4 & (−5, −16, −7, 12, 0, −2) \\ \sqrt{5} & −3 & (−5, −12, −3, 7, 0, −1) \\ 7 & 5 & (24, 37, 5, −27, 2, 5) \\ 11A & −4 & (23, 35, 7, −24, 1, 4) \\ 11B & −3 & (1, 11, 9, −8, −1, 1) \\ 19A & 3 & (−27, −24, 3, 16, −2, −3) \\ 19B & 4 & (37, 38, −1, −24) \\ 29A & 0 & (−5, 2, 1, 0, 0, 0) \\ 29B & 0 & (−61, −72, −1, 46, −4, −8) \\ 31A & 10 & (−15, −24, −5, 14, 0, −2) \\ 31B & −1 & (10, 22, 10, −18, 0, 3) \\ 41A & −4 & (39, 56, 7, −36, 2, 6) \\ 41B & −10 & (34, 24, −8, −12, 2, 2) \\ 59A & −5 & (19, 11, −5, −6, 1, 1) \\ 59B & −2 & (−22, −22, −4, 14, 0, −2) \\ 61A & −1 & (−19, −41, −16, 31, 0, −5) \\ 61B & −6 & (−71, −78, 1, 48, −4, −8) \\ 71A & 0 & (24, 8, −14, 0, 2, 0) \\ 71B & −15 & (−47, −62, −9, 42, −2, −7) \\ 79A & 0 & (−3, 19, 13, −14, −1, 2) \\ 79B & −16 & (27, 42, 1, −24, 2, 4) \\ 89A & 3 & (19, 36, 7, −27, 2, 5) \\ 89B & 15 & (2, −17, −15, 11, 2, −1) \end{aligned} \begin{matrix}409A : x^{4} − x^{3} − 8x^{2} + 4x + 8 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ −3 & 3 − ω_{5} & (0, 1, 0, 0) \\ −4 & 3 & (0, −3, −1, 1) \\ −1 & −2 + 2ω_{5} & (0, 3, 2, −2) \\ 2 & −2 + 8ω_{5} & (6, 8, 0, −4) \\ −1 & −4 + 2ω_{5} & (−2, −5, 0, 2) \\ −1 & 1 − 2ω_{5} & (6, 2, −3, 0) \\ 0 & −3 + 2ω_{5} & (2, −8, −4, 5) \\ −4 & 2 − 6ω_{5} & (0, −2, 2, 0) \\ −4 & −6 & (8, 4, 0, −2) \\ −3 & −2 + 4ω_{5} & (−1, −2, 0, 0) \\ −1 & −3 + 2ω_{5} & (−2, −3, −2, 3) \\ −1 & 2 & (−2, −5, −4, 4) \\ 8 & 7 − 4ω_{5} & (−10, 1, 3, −2) \\ −6 & −1 + 4ω_{5} & (−10, 0, 6, −3) \\ 0 & −7 + 2ω_{5} & ([8, 5, −1, −1) \\ 12 & −8ω_{5} & (6, 10, 0, −4) \\ 6 & 2 & (3, 5, 2, −4) \\ 5 & −4 + 6ω_{5} & (−6, 9, 4, −4) \\ −14 & −12 + 4ω_{5} & (5, 3, −2, 0) \\ 15 & 3 − 2ω_{5} & (4, 6, −3, −1) \\ −11 & −8 + 12ω_{5} & (7, 2, −4, 0) \\ 1 & 2 + 4ω_{5} & (−6, 3, 2, 0) \\ 6 & 13 − 8ω_{5} & (−14, 3, 11, −6) \\ 1 & 7 + 4ω_{5} & (6, −7, −8, 7) \end{matrix}

The integral basis for the field ℚ(α), where α is a root of x^{6} − 15x^{4} +

12x^{3} + 39x^{2} − 24x − 29 is

\{1, α, α^{2}, \frac{1}{2}(α^{3} + α^{2} + α + 1), \frac{1}{2}(α^{4} + 1), \frac{1}{4}(α^{5} + α^{4} − α − 1)\}.

The integral basis for the field ℚ(α), where α is a root of x^{4} − x^{3} − 8x^{2} +

4x + 8, is

\{1, α, \frac{1}{2}(α^{2} + α), \frac{1}{4}(α^{3} + α^{2} + 2α)\}.

\begin{aligned}& 419A : x^{4} − 6x^{2} + 2x + 5 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 2 & −1 & −3 & (0, 1, 0, 0) \\ 3 & −4 & −4 & (−1, 0, 1, 0) \\ \sqrt{5} & −4 & 0 & (3, 0, −1, 0) \\ 7 & 6 & 6 & (10, 8, −4, −2) \\ 11A & −1 & −3 & (4, 7, −2, −2) \\ 11B & 3 & −5 & (−3, −8, 2, 2) \\ 19A & −6 & 2 & (2, −2, 0, 0) \\ 19B & −6 & 2 & (−4, −5, 2, 1) \\ 29A & −4 & 0 & (−4, −3, 2, 1) \\ 29B & 6 & −10 & (12, 1, −4, −1) \\ 31A & 10 & 2 & (2, 10, −2, −2) \\ 31B & −10 & 10 & (1, 0, −1, 0) \\ 41A & 4 & 0 & (−17, −14, 7, 4) \\ 41B & 2 & −2 & (−12, 2, 4, 0) \\ 59A & −12 & −12 & (13, 14, −5, −4) \\ 59B & −5 & −13 & (−6, 4, 1, 0) \\ 61A & 10 & −10 & (4, 6, −4, −2) \\ 61B & −1 & 13 & (2, 1, −2, 0) \\ 71A & −14 & 6 & (−1, −4, 1, 2) \\ 71B & −14 & 6 & (−10, 2, 4, 0) \\ 79A & 6 & 10 & (8, −1, −2, −1) \\ 79B & −5 & −15 & (6, −14, 0, 3) \\ 89A & −10 & −14 & (−4, 3, 2, −1) \\ 89B & 15 & 7 & (1, 20, −2, −4) \end{aligned} 421A : x^{8} − 2x^{7} − 23x^{6} + 46x^{5}+ ; 148x^{4} − 312x^{3} − 176x^{2} + 576x − 256 ; a_{𝔭}(f_{1}) ; (0, 1, 0, 0, 0, 0, 0, 0) ; (6, −16, −1, 14, 0, −7, −1, 2) ; (13, −19, −7, 16, 3, −9, −2, 3) ; (46, −73, −29, 65, 14, −35, −8, 11) ; (−39, 66, 28, −61, −14, 34, 8, −11) ; (12, −5, −13, 5, 8, −3, −2, 1) ; (−36, 42, 24, −30, −12, 14, 4, −4) ; (8, −18, −10, 16, 7, −9, −3, 3) ; (62, −78, −48, 70, 24, −38, −10, 12) ; (40, −73, −24, 62, 10, −34, −7, 11) ; (−72, 106, 50, −92, −24, 50, 12, −16) ; (−80, 134, 56, −122, −28, 68, 16, −22) ; (−56, 96, 42, −90, −22, 50, 12, −16) ; (117, −179, −79, 162, 38, −90, −21, 29) ; (−62, 76, 48, −64, −24, 36, 10, −12) ; (−91, 136, 64, −121, −32, 66, 16, −21) ; (52, −74, −38, 72, 18, −42, −10, 14) ; (138, −204, −94, 178, 45, −97, −23, 31) ; (58, −70, −46, 56, 25, −29, −9, 9) ; (−22, 28, 14, −30, −6, 18, 4, −6) ; (63, −109, −55, 102, 30, −58, −15, 19) ; (−156, 230, 109, −198, −56, 107, 27, −34) ; (−32, 60, 18, −50, −10, 26, 6, −8) ; (64, −83, −47, 75, 22, −41, −10, 13)

For ℚ(α) ≅ ℚ[x]/(x^{8} − 2x^{7} − 23x^{6} + 46x^{5} + 148x^{4} − 312x^{3} − 176x^{2} + 576x − 256),

the chosen integral basis is

{1, α, α^{2}, 1/2(α^{3} + α), 1/4(α^{4} + α^{2}), 1/8(α^{5} + α^{3} + 4α),

1/16(α^{6} + α^{4} − 4α^{2}), 1/64(α^{7} + 2α^{6} + α^{5} + 2α^{4} + 12α^{3} + 4α^{2} + 16α).

\begin{aligned}& & 431A : x^{3} + 3x^{2} − 9x − 23 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) & a_{𝔭}(f_{4}) & a_{𝔭}(f_{5}) \\ 2 & 1 & 2 & −1 & −2 & (0, 1, 0) \\ 3 & 2 & −5 & −2 & −1 & (2, 0, 0) \\ \sqrt{5} & 2 & 0 & 3 & −2 & (−6, 0, 1) \\ 7 & −10 & −4 & 5 & −4 & (−5, 1, 1) \\ 11A & 0 & 6 & 0 & −6 & (1, −1, 0) \\ 11B & −4 & 0 & 6 & −2 & (12, 0, −2) \\ 19A & 4 & −1 & −4 & −5 & (8, −2, −2) \\ 19B & −4 & 5 & −1 & 5 & (−12, −1, 2) \\ 29A & −2 & 9 & −3 & 5 & (7, 0, −2) \\ 29B & −2 & −9 & −3 & −1 & (5, 0, 0) \\ 31A & 0 & −4 & 5 & 0 & (−9, 0, 2) \\ 31B & 4 & 2 & −1 & 4 & (15, −1, −3) \\ 41A & −6 & 9 & −6 & −3 & (−12, 2, 4) \\ 41B & 2 & −9 & 12 & −5 & (−14, 0, 2) \\ 59A & 8 & −12 & −3 & 4 & (−3, 3, 1) \\ 59B & 4 & −12 & 6 & 2 & (−14, 2, 2) \\ 61A & −6 & 8 & −1 & −12 & (17, 1, −3) \\ 61B & 2 & −4 & 2 & 2 & (7, −3, −2) \\ 71A & 8 & 3 & −12 & −5 & (12, −2, −2) \\ 71B & 8 & −6 & −12 & −2 & (8, 0, 0) \\ 79A & 8 & −4 & 2 & 14 & (−8, 2, 0) \\ 79B & 0 & −1 & −10 & 3 & (14, 2, −2) \\ 89A & 10 & 15 & −15 & −1 & (10, −3, −4) \\ 89B & 2 & −6 & −12 & −8 & (−16, 0, 4) \end{aligned} \begin{matrix}2 · 109A : x^{3} − x^{2} − 20x − 25, \\ x^{3} − x^{2} − 8x − 5 \\ a_{𝔭}(f_{1}) : α_{1} & a_{𝔭}(f_{2}) : α_{2} \\ (−1, 0, 0) & (1, 0, 0) \\ (0, 1, 0) & (0, 1, 0) \\ (−2, −1, 1) & (6, 2, −1) \\ (4, 1, −1) & (0, 2, −1) \\ (3, 0, −1) & (−3, −1, 1) \\ (10, 2, −4) & (2, −2, 0) \\ (−14, −2, 6) & (−10, −4, 2) \\ (−6, −1, 2) & (−10, −5, 2) \\ (10, 2, −6) & (−10, −4, 2) \\ (−6, 0, 2) & (10, 6, −2) \\ (−11, −2, 5) & (−3, −3, 1) \\ (20, 3, −7) & (−8, −2, 1) \\ (4, 2, −2) & (12, 4, −2) \\ (13, 2, −7) & (17, 9, −3) \\ (−4, 1, 0) & (0, −1, 0) \\ (8, −2, −2) & (0, 4, −2) \\ (17, 0, −5) & (−13, 3, 1) \\ (−16, −3, 6) & (−8, −7, 2) \\ (−14, −4, 4) & (−18, −8, 4) \\ (−32, −4, 14) & (12, 2, −2) \\ (−24, −4, 8) & (20, 4, −4) \\ (−3, 2, −1) & (5, 3, −1) \\ (6, 3, −6) & (10, 1, −2) \\ (22, 5, −11) & (−10, −4, 1) \end{matrix}

The integral basis for the field ℚ(α), where α is a root of the polynomial

x^{3} + 3x^{2} − 9x − 23 is

{1, α, 1/2(α^{2} + 1)}.

The integral basis for the field ℚ(α_{1}), where α_{1} is a root of the polyno-

mial x^{3} − x^{2} − 20x − 25, is

{1, α_{1}, 1/5(α_{1}^{2} − α_{1})}.

\begin{aligned}& 439A : x^{3} − x^{2} − 9x + 5 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 2 & 2 − ω_{5} & −2 − ω_{5} & (0, 1, 0) \\ 3 & 2 + ω_{5} & −4 + ω_{5} & (−4, −1, 1) \\ \sqrt{5} & −2 + ω_{5} & −2 + ω_{5} & (2, 0, 0) \\ 7 & 2 − 5ω_{5} & 2 + 7ω_{5} & (0, −2, 0) \\ 11A & 1 & 1 − 4ω_{5} & (−1, −1, 0) \\ 11B & 1 & −3 + 4ω_{5} & (6, 0, −2) \\ 19A & −1 + 4ω_{5} & −3 − 2ω_{5} & (4, 0, 0) \\ 19B & 2 − 6ω_{5} & −2 − 2ω_{5} & (6, −1, −1) \\ 29A & 1 + 5ω_{5} & 1 − ω_{5} & (4, 0, −2) \\ 29B & −3 & −1 − 2ω_{5} & (4, −2, 0) \\ 31A & −1 & −1 − 4ω_{5} & (−6, 0, 2) \\ 31B & 2 + ω_{5} & 5ω_{5} & (−10, 0, 2) \\ 41A & −3 − 4ω_{5} & −1 − 2ω_{5} & (2, 1, −1) \\ 41B & −3 + 7ω_{5} & 9 − 5ω_{5} & (−4, 2, 0) \\ 59A & 7 − 3ω_{5} & −1 + 7ω_{5} & (−10, 0, 2) \\ 59B & −7 + 7ω_{5} & −7 + 7ω_{5} & (−6, 0, 2) \\ 61A & 1 − 2ω_{5} & 7 − 4ω_{5} & (−2, 1, 1) \\ 61B & 3 + 6ω_{5} & −11 + 4ω_{5} & (−12, 1, 1) \\ 71A & −11 − 2ω_{5} & 7 − 4ω_{5} & (−14, 2, 4) \\ 71B & −5 & −11 + 6ω_{5} & (0, 1, 1) \\ 79A & −9 + 7ω_{5} & −5 + ω_{5} & (16, 3, −3) \\ 79B & 2 − 4ω_{5} & −6 + 12ω_{5} & (3, 3, 0) \\ 89A & −2 − 5ω_{5} & −6 − 5ω_{5} & (17, 1, −6) \\ 89B & −5 − 6ω_{5} & 5 − 4ω_{5} & (10, 6, −2) \end{aligned} \begin{aligned}& 3 · 7 : x^{3} − x^{2} − 13x + 5 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) & a_{𝔭}(f_{4}) & a_{𝔭}(f_{5}) \\ 3 & −3 & 1 & 1 & (−1, 2, 0) \\ −1 & 1 & 1 & 1 & (−1, 0, 0) \\ 2 & −2 & 2 & 2 & (2, −2, 0) \\ −1 & 1 & 1 & 1 & (−1, 0, 0) \\ −6 & 4 & −4 & 4 & (−3, 0, 0) \\ −6 & 4 & 4 & −4 & (−3, 0, 0) \\ 6 & 4 & −8 & 0 & (2, 0, −2) \\ 6 & 4 & 0 & −8 & (2, 0, −2) \\ 2 & −2 & 6 & −10 & (2, −4, 0) \\ 2 & −2 & −10 & 6 & (2, −4, 0) \\ −10 & 0 & −4 & 4 & (2, 4, −2) \\ −10 & 0 & 4 & −4 & (2, 4, −2) \\ 2 & 2 & 6 & 6 & (−6, −2, 4) \\ 2 & 2 & 6 & 6 & (−6, −2, 4) \\ 8 & 12 & 12 & −4 & (−16, 4, 4) \\ 8 & 12 & −4 & 12 & (−16, 4, 4) \\ −2 & −2 & −10 & 6 & (6, 0, −4) \\ −2 & −2 & 6 & −10 & (6, 0, −4) \\ 10 & 0 & 0 & −8 & (−8, 0, 0) \\ 10 & 0 & −8 & 0 & (−8, 0, 0) \\ −4 & −16 & 8 & −8 & (4, 0, −4) \\ −4 & −16 & −8 & 8 & (4, 0, −4) \\ −6 & −14 & 6 & 6 & (−6, 2, 0) \\ −6 & −14 & 6 & 6 & (−6, 2, 0) \end{aligned}

The integral basis for the field ℚ(α), where α is a root of the polynomial

x^{3} − x^{2} − 9x + 5 is

{1, α, 1/2(α^{2} + 1)}.

The integral basis for the field ℚ(α), where α is a root of the polynomial

x^{3} − x^{2} − 13x + 5, is

{1, 1/2(α + 1), 1/4(α^{2} − 1)}.

\begin{aligned}& & \sqrt{5} · 89B : x^{3} − x^{2} − 12x + 4 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{3}) & a_{𝔭}(f_{3}) & a_{𝔭}(f_{4}) \\ 2 & 1 & −3 & 2 & (0, 1, 0) \\ 3 & 2 & −2 & 2 − ω_{13} & (−4, −1, 1) \\ \sqrt{5} & 1 & −1 & −1 & (1, 0, 0) \\ 7 & −6 & 6 & 7 − 5ω_{13} & (−3, −2, 1) \\ 11A & 0 & 0 & 2 − 2ω_{13} & (6, 0, −2) \\ 11B & 0 & −4 & −2 + 2ω_{13} & (6, 0, −2) \\ 19A & −8 & 0 & −1 + 3ω_{13} & (1, 0, 1) \\ 19B & 4 & −4 & 2ω_{13} & (−8, −2, 2) \\ 29A & −10 & −10 & 3 − 3ω_{13} & (5, 0, −1) \\ 29B & −2 & −2 & 0 & (8, 2, −4) \\ 31A & 4 & 0 & −2 + 4ω_{13} & (−2, −2, 0) \\ 31B & 0 & −8 & −1 − 3ω_{13} & (−9, −2, 5) \\ 41A & 10 & 2 & −8 − ω_{13} & (−10, −1, 3) \\ 41B & 2 & 6 & −6 & (10, 4, −4) \\ 59A & 4 & −4 & −8 + 2ω_{13} & (0, 2, −2) \\ 59B & 0 & 12 & 6 − 3ω_{13} & (8, 3, −5) \\ 61A & −2 & −2 & 3 − 7ω_{13} & (−1, 0, 1) \\ 61B & 14 & −6 & 2 & (−14, 0, 4) \\ 71A & −16 & 0 & −6 + 3ω_{13} & (8, −1, −1) \\ 71B & 8 & 16 & −12 + 3ω_{13} & (−2, 1, −1) \\ 79A & −4 & 8 & 3 + 5ω_{13} & (1, 2, −1) \\ 79B & −12 & 4 & 8 − 3ω_{13} & (−6, −3, 3) \\ 89A & −6 & −6 & 8 + ω_{13} & (−18, −5, 7) \\ 89B & 1 & 1 & −1 & (−1, 0, 0) \end{aligned} \begin{matrix}11A · 41B : x^{3} − x^{2} − 9x + 5, \\ x^{3} + x^{2} − 9x + 3 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) : α_{1} & a_{𝔭}(f_{3}) : α_{2} \\ 3 & (0, 1, 0) & (0, 1, 0) \\ 4 & (3, 0, −1) & (3, −2, −1) \\ −2 & (1, −1, 0) & (1, 1, 0) \\ −10 & (0, 2, 0) & (−12, 2, 4) \\ −1 & (1, 0, 0) & (1, 0, 0) \\ −6 & (−1, 0, 1) & (5, −2, −1) \\ 6 & (−3, 0, 1) & (−5, 0, 1) \\ 4 & (6, 0, −2) & (−6, 0, 2) \\ 0 & (3, −2, −1) & (−5, 2, 1) \\ 2 & (−5, 1, 0) & (13, −3, −4) \\ −10 & (5, 2, −1) & (−3, 2, 1) \\ 4 & (−4, −2, 2) & (8, −2, −2) \\ −8 & (5, 0, −1) & (3, −4, −1) \\ 1 & (−1, 0, 0) & (−1, 0, 0) \\ 0 & (−6, −4, 2) & (−10, 4, 2) \\ 4 & (1, −1, −2) & (19, −5, −6) \\ −14 & (14, 0, −4) & (−10, 4, 4) \\ −4 & (−17, 2, 3) & (13, −2, −1) \\ 2 & (5, 4, −1) & (−7, 2, 3) \\ −8 & (−2, 0, −2) & (14, 0, −2) \\ 14 & (3, 0, −1) & (−17, 4, 5) \\ −8 & (−10, 2, 0) & (10, 2, 0) \\ 10 & (−12, −2, 4) & (−16, 6, 4) \\ 12 & (−15, 0, 5) & (−1, −2, −1) \end{matrix}

The chosen integral basis for the field ℚ(α), where α is a root of x^{3} −

x^{2} − 12x + 4, is {1, α, 1/4(α^{2} + α + 2)}.

The integral basis for the field ℚ(α_{1}), where α_{1} is a root of x^{3} −x^{2} −9x+5,

is {1, α_{1}, 1/2(α_{1}^{2} + 1)}.

The integral basis for the field ℚ(α_{2}), where α_{2} is a root of x^{3} +x^{2} −9x+3,

is {1, α_{2}, 1/2(α_{2}^{2} + 1)}.

\begin{aligned}& 461A : x^{4} − 4x^{3} − 12x^{2} + 48x − 16 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 2 & −2 − ω_{5} & −1 − ω_{5} & (−1, 3, 1, −1) \\ 3 & −1 & −3 & (0, 2, 0, 0) \\ \sqrt{5} & 3 − 2ω_{5} & −3 + 2ω_{5} & (3, 0, −1, 0) \\ 7 & 5 + 3ω_{5} & 8 − 9ω_{5} & (−12, 22, 4, −6) \\ 11A & 3ω_{5} & −1 − 3ω_{5} & (6, −8, −2, 2) \\ 11B & 5 − ω_{5} & −4 − ω_{5} & (−5, −1, 2, 0) \\ 19A & −3 + 3ω_{5} & 4 − 3ω_{5} & (4, −6, −2, 2) \\ 19B & −3 + 3ω_{5} & −4 + 5ω_{5} & (4, −6, −2, 2) \\ 29A & −1 + 5ω_{5} & 4 − 3ω_{5} & (−6, 8, 2, −2) \\ 29B & 5 − 2ω_{5} & −1 + 6ω_{5} & (−10, 8, 3, −2) \\ 31A & −3 + 6ω_{5} & −3 + 2ω_{5} & (4, −6, −1, 1) \\ 31B & 1 − 6ω_{5} & −1 + 6ω_{5} & (−4, −8, 2, 2) \\ 41A & −7 + 4ω_{5} & −1 + 4ω_{5} & (8, −8, −2, 2) \\ 41B & 2 − ω_{5} & −9 − ω_{5} & (10, −4, −4, 2) \\ 59A & −6 & −2 + 8ω_{5} & (4, 2, −1, −1) \\ 59B & 2 − 2ω_{5} & 8 − 6ω_{5} & (−11, 13, 4, −4) \\ 61A & 6 + 3ω_{5} & −3 − 3ω_{5} & (−14, 16, 5, −4) \\ 61B & 6 − 8ω_{5} & −2 − 4ω_{5} & (16, −18, −7, 4) \\ 71A & −4 + 8ω_{5} & −12 & (−12, 22, 6, −6) \\ 71B & 2 − 10ω_{5} & 4 − 6ω_{5} & (−6, 6, 3, −1) \\ 79A & −9 + 3ω_{5} & 6 − 11ω_{5} & (14, −12, −6, 4) \\ 79B & 1 + 6ω_{5} & 5 + 2ω_{5} & (−13, 9, 4, −4) \\ 89A & −16 − ω_{5} & −3 − 5ω_{5} & (10, −2, −2, 0) \\ 89B & −9 & −7 + 8ω_{5} & (−2, 16, 2, −4) \end{aligned} \begin{matrix}479A : x^{4} + x^{3} − 8x^{2} − 11x + 1 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) & a_{𝔭}(f_{3}) \\ 1 & −1 − ω_{5} & (0, 1, 0, 0) \\ 2 & −3 & (2, −9, −3, 4) \\ 2 & −3 + ω_{5} & (−1, 8, 3, −4) \\ 10 & −7 + 2ω_{5} & (4, 9, 1, −4) \\ −2 & −3 + 4ω_{5} & (0, −5, −1, 2) \\ 0 & −1 − 2ω_{5} & (4, 1, −1, 0) \\ 2 & −3 − 3ω_{5} & (1, −6, −1, 2) \\ −4 & 1 − ω_{5} & (3, 16, 5, −8) \\ −6 & −2 + 6ω_{5} & (8, −6, −4, 4) \\ −6 & 1 − 7ω_{5} & (−3, −10, −1, 4) \\ 8 & 4 − 3ω_{5} & (−3, −25, −8, 12) \\ −8 & −2 − ω_{5} & (7, −7, −4, 4) \\ 6 & −3 + 9ω_{5} & (−5, 8, 3, −4) \\ 4 & 5 − 3ω_{5} & (−1, 4, 1, −2) \\ 2 & −11 + 2ω_{5} & (6, −11, −5, 6) \\ 12 & 1 + 6ω_{5} & (−10, −9, −1, 4) \\ −8 & 4 − 2ω_{5} & (−2, −10, −4, 6) \\ −12 & 5 − 2ω_{5} & (4, 23, 5, −10) \\ −10 & 2 − 8ω_{5} & (−10, 28, 12, −14) \\ 12 & 1 + 3ω_{5} & (−15, 6, 5, −4) \\ 16 & 7 − 2ω_{5} & (−4, 25, 7, −12) \\ 0 & −1 − 5ω_{5} & (−9, −2, 3, 0) \\ −10 & −5 + 2ω_{5} & (10, 39, 9, −16) \\ −10 & −9 + 14ω_{5} & (6, −33, −11, 16) \end{matrix}

The chosen integral basis for the field ℚ(α), where α is a roots of x^{4} −

4x^{3} − 12x^{2} + 48x − 16, is

{1, 1/2α, 1/4α^{2}, 1/8α^{3}}.

The chosen integral basis for the field ℚ(α), where α is a roots of x^{4} +

x^{3} − 8x^{2} − 11x + 1, is

{1, α, α^{2}, 1/4(α^{3} + 2α^{2} + 2α − 1)}.

\begin{aligned}& 491A : x^{6} − x^{5} − 13x^{4} \\ & +16x^{3} + 39x^{2} − 63x + 17 \\ 𝔭 & a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) \\ 2 & −1 − ω_{5} & (0, 1, 0, 0, 0, 0) \\ 3 & −1 − 2ω_{5} & (−7, 7, 1, −1, 0, 0) \\ \sqrt{5} & −3 + 2ω_{5} & (2, 3, −4, −1, 1, 0) \\ 7 & 5 + 3ω_{5} & (−3, 23, −5, −12, 1, 2) \\ 11A & −2 − ω_{5} & (4, −7, 0, 1, 0, 0) \\ 11B & −5 & (5, −4, 3, 1, −1, 0) \\ 19A & 1 + 2ω_{5} & (−17, 33, 1, −13, 0, 2) \\ 19B & −2 + 3ω_{5} & (18, −41, 6, 15, −2, −2) \\ 29A & −6 + 7ω_{5} & (3, −16, 4, 7, −1, −1) \\ 29B & 3 − 2ω_{5} & (11, −21, −1, 11, 0, −2) \\ 31A & −1 − 5ω_{5} & (−1, 6, −7, −2, 2, 0) \\ 31B & 4 − 8ω_{5} & (−7, 16, −3, −7, 1, 1) \\ 41A & −1 + 7ω_{5} & (−13, 28, 1, −12, 0, 2) \\ 41B & −3 + 3ω_{5} & (−17, 14, 3, −2, 0, 0) \\ 59A & 2 − 6ω_{5} & (−20, 28, 2, −12, 0, 2) \\ 59B & 1 + ω_{5} & (−11, 14, 1, −10, 0, 2) \\ 61A & −14 + ω_{5} & (2, 19, −2, −11, 0, 2) \\ 61B & 3 + 4ω_{5} & (17, −33, −1, 13, 0, −2) \\ 71A & 7 − 7ω_{5} & (33, −54, −3, 24, 0, −4) \\ 71B & −6 − 4ω_{5} & (8, −17, 10, 4, −3, 0) \\ 79A & −6 − 6ω_{5} & (−18, 6, 10, 0, −2, 0) \\ 79B & 7ω_{5} & (11, −14, 2, 7, −1, −1) \\ 89A & 1 − 4ω_{5} & (−30, 69, 4, −30, −1, 5) \\ 89B & 10 − ω_{5} & (10, −31, 0, 13, 0, −2) \end{aligned} \begin{matrix}4 · 31A : x^{4} + 12x^{3} \\ −24x^{2} − 80x + 80 \\ a_{𝔭}(f_{1}) & a_{𝔭}(f_{2}) \\ 0 & (0, 0, 0, 0) \\ −1 + ω_{3} & (0, 1, 0, 0) \\ 1 − ω_{3} & (20, −11, −2, 6) \\ −4 − 2ω_{3} & (−6, 6, 0, −4) \\ 1 + ω_{3} & (2, −1, 0, 0) \\ 2 & (−20, 14, 2, −8) \\ 5 + ω_{3} & (−18, 5, 2, −2) \\ −2 − 2ω_{3} & (−32, 14, 4, −8) \\ −2 + 4ω_{3} & (6, −8, 0, 4) \\ −2 + 2ω_{3} & (10, 0, −2, 0) \\ −1 & (1, 0, 0, 0) \\ 5 − 3ω_{3} & (10, −3, −2, 2) \\ 6ω_{3} & (2, −2, 0, 0) \\ −6ω_{3} & (−30, 14, 4, −8) \\ −6 + 2ω_{3} & (32, −14, −4, 8) \\ −4 & (0, 0, 0, 0) \\ −3 + 3ω_{3} & (40, −21, −4, 12) \\ −2 + 4ω_{3} & (−30, 16, 4, −8) \\ 9 − 3ω_{3} & (30, −21, −4, 12) \\ −2 + 6ω_{3} & (28, −6, −4, 4) \\ −6 − 2ω_{3} & (44, −30, −4, 16) \\ 4 − 4ω_{3} & (−48, 24, 4, −12) \\ −6 − 4ω_{3} & (70, −36, −8, 20) \\ −8 − 6ω_{3} & (6, −10, 0, 4) \end{matrix}

For ℚ(α) ≅ ℚ[x]/(x^{6} − x^{5} − 13x^{4} + 16x^{3} + 39x^{2} − 63x + 17), the chosen integral

basis is

{1, α, α^{2}, α^{3}, 1/2(α^{4} + α^{3} + α + 1), 1/2(α^{5} + α^{3} + α^{2} + 1)}.

For ℚ(α) ≅ ℚ[x]/(x^{4} + 12x^{3} − 24x^{2} − 80x + 80) the chosen integral basis:

{1, α, 1/2α^{2}, 1/10(α^{3} − α^{2} + 4α)}.

8. Motives.

By a gp search, we found the list of all curves of prime conductor of norm less than 100. We would like to thank Prof. N. Elkies for his valuable help in implementing this search. We have only listed one curve for each prime as one gets the other one by Galois conjugation. For each curve E/ℚ(\sqrt{5}) of conductor 𝔠, we have checked that all the a_{𝔭}(E) match up with the Fourier coefficients of a modular form of level 𝔠 given in the table. We would like to complete the list of those curves in a near future along with studing their modularity by making use of some of the results in Skinner and Wiles [50, 51]. In fact, we already have the modularity of the curves listed below thanks to a result in [51]. We begin by recalling this theorem.

Definition 1.45. Let ρ̄: Gal(F̄/F) → GL_{2}(\overline{𝔽}_{p}) be a Galois representation, and let 𝔭|p be a prime of F. We say that ρ̄ is distinguished at 𝔭 if the semisimplification of the restriction of ρ̄ to D_{𝔭}, the decomposition group at 𝔭, is isomorphic to θ_{1} ⊕ θ_{2}, where θ_{1} and θ_{2} are two distincts characters from Gal(F̄/F) to \overline{𝔽}_{p}^{×}.

Proposition 1.2. in Ellenberg [22], says that the mod p Galois representations associated to the curves listed below are 𝔭–distinguished for all 𝔭 not dividing the conductor. This is also a consequence of the corollary to Proposition 11 in Serre [44].

Here is the theorem we need from Skinner and Wiles [51] (see [22, Theorem 3.1]).

Theorem 1.46. Let F be a totally real number field, let p > 2 be a rational prime, let L be a finite extension of ℚ_{p}, and let

ρ : Gal(F̄/F) → GL_{2}(L)

be a continuous irreducible Galois representation ramified at only finitely many primes. Suppose

• det ρ = ψχ_{p}^{k−1}, where χ_{p} is the cyclotomic character and ψ is a finite order character;

• ρ is ordinary in the sense that, for each prime 𝔭 dividing p,

ρ|I_{𝔭} ≅ (\begin{matrix}χ_{p}^{k−1} & ∗ \\ 0 & 1 \end{matrix});

• ρ is distinguished at 𝔭 for each prime 𝔭 dividing p;

• there exists an ordinary modular Galois representation ρ′ and an isomorphism between the mod p representations ρ̄ and ρ̄′.

Then ρ is modular.

Proof. See [51]. □

For the rest of this section, we let F = ℚ(\sqrt{5}). Let E be an ellipitc curve defined over F. For any prime ℓ ≥ 3, let

ρ̄_{E, ℓ} : Gal(F̄/F) → GL_{2}(𝔽_{ℓ})

be the mod ℓ representation attached to E. The proof of Serre [45, Proposition 1] carries on to give the following lemma.

Lemma 1.47. The image of ρ̄_{E, 3} is either GL_{2}(𝔽_{3}) or contained in a Borel subgroup; the latter case happens if and only if E or a 3-isogenous curve E′/F to it has a 3–torsion point defined over F.

82 1. COMPUTING HILBERT MODULAR FORMS.

Proof. Going through the proof of Proposition 1 in Serre [45], one sees that the only thing we need to check is that there can’t be a Galois extension K of F = ℚ(\sqrt{5}) such that Gal(K/F) = D_{4} or a subgroup of index 2 in D_{4} and K ramifies only at 3. We recall that D_{4}/ℤ_{2} ≅ (ℤ/2ℤ)^{2}. But, there can’t be an abelian extension of F of degree 4 that is only ramified at 3. □

Let E/ℚ(\sqrt{5}) be any of the curve listed below, let ρ_{E, 3} be the associated 3–adic Galois representation and ρ̄_{E, 3} its reduction mod 3. By inspecting the listed of the a_{𝔭}, we see that, except for the curves of conductor of norm 49, 71 or 89, the curve E does not have a 3–torsion point; neither can it be 3–isogenous to a curve E′/F that has a 3-torsion point defined over F. Therefore by Lemma 1.47, ρ̄_{E, 3} is irreducible, and so is ρ_{E, 3}. As we already mentionned, E is distinguished at 3. It is also ordinary at 3, since it has good reduction at 𝔭 = 3 and a_{3}(E) is not divisible by 3; this follows from results in Chap. V, VIII and IX in Silverman [49] (on elliptic curves over finite field and reduction of elliptic curves over local or global fields.). The ordinariness in this sense is equivalent to the one in Theorem 1.46 by the theory of finite flat commutative group schemes. The determinant of ρ_{E, 3} being cyclotomic is also a well–known fact. So, we only need check that ρ̄_{E, 3} is modular. But since GL_{2}(𝔽_{3}) is solvable, this follows from Langlands–Tunnell (see [36, Chap. I] and [52], see also Serre [46]). Therefore, E is modular.

Remark 1.48. The curve E/F with conductor of norm 41 in our list has a 7–torsion point defined over F. So we could have used instead the Main theorem of Skinner–Wiles [50] to deduce its modularity. This has the advantage of avoiding to verify the modularity of the residual representation ρ̄_{E, 7}. We can also use the same result to conclude the modularity for the rest of the curves in our list which might have a reducible mod 3 representation.

\begin{matrix}𝔠 & a_{1} & a_{2} & a_{3} & a_{4} & a_{6} \\ 31 & 1 & −1 − ω_{5} & ω_{5} & 0 & 0 \\ 41 & 0 & −ω_{5} & ω_{5} & 0 & 0 \\ 7 & 0 & ω_{5} & 1 & 1 & 0 \\ 71 & 1 + ω_{5} & −1 + ω_{5} & 1 & 0 & 0 \\ 79 & 1 + ω_{5} & −1 + ω_{5} & ω_{5} & 0 & 0 \\ 89 & ω_{5} & −ω_{5} & 1 & −1 & 0 \end{matrix}

9. Listing of Programs.

The first program computes the Hecke eigenvalues for a given level,

and second one the quaternions needed to calculate the Hecke action.

\\THIS COMUPTE ALL HECKE EIGENVALUES FOR A GIVEN LEVEL.

\\Initial values.
w=quadgen(5);
field=nfinit(z^2-z-1);
fieldembed(x)=Mod(real(x)+imag(x)*z, z^2-z-1);
level=fieldembed(6+w);
levelf=idealfactor(field, level);
level0=[]; res=idealfactor(field, level);\
for(j=1, matsize(res)[1], res0=[res[j,][1][1],res[j,][2]];\
    level0=concat(level0, [res0])); print(level0);
level1=length(level0);
read(icosiangroup1);

\\Working with quaternions; we represent quaternions by vectors [x,y,w,z].
hnorm(q)=q[1]^2+q[2]^2+q[3]^2+q[4]^2;
hconj(q)=[q[1],-q[2],-q[3],-q[4]];
hmult(a,b)=[a[1]*b[1]-a[2]*b[2]-a[3]*b[3]-a[4]*b[4],\
           a[1]*b[2]+a[2]*b[1]+a[3]*b[4]-a[4]*b[3],\
           a[1]*b[3]+a[3]*b[1]-a[2]*b[4]+a[4]*b[2],\
           a[1]*b[4]+a[4]*b[1]+a[2]*b[3]-a[3]*b[2] ];
hinv(q)=hconj(q)/hnorm(q);

\\This function returns the reduction of a matrix modulo level with
\\coefficients in O_K.
matredmod(u,j,res0)=res0=matrix(matsize(u)[1], matsize(u)[2]);\
    for(p1=1, matsize(u)[1], for(q1=1, matsize(u)[2],\
           res0[p1,q1]=u[p1,q1]%(level0[j][1]^level0[j][2]))); res0;
\\Be carefull in using the function vectredmod: one might have some
\\problems when the denominator is divisible by 2.
vectredmod(v, l, res1)=res1=vector(length(v));\
        for(p1=1, length(v), res2=v[p1]%(level0[l][1]^level0[l][2]);\
             res2=nfeltreduce(field, fieldembed(res2),\
             idealpow(field, levelf[l,][1], levelf[l,][2]));\
             res1[p1]=res2[1]+res2[2]*w); res1;

\\Here is the quotient ring O/level.
Quotring=vector(length(level0));\
  for(j=1, length(level0), res=[]; card=level0[j][1]^level0[j][2];\
      if(kronecker(level0[j][1],5)==-1,\
             for(i=0, card-1, for(k=0, card-1, res=concat(res, i+k*w))),\
             for(i=0, card-1, res=concat(res, i))); Quotring[j]=res);\
              print(Quotring);

Proj0=vector(level1);\
for(j=1, level1, res=[]; for(k=1, length(Quotring[j]),\
          res=concat(res, [[1, Quotring[j][k]]]));\
  if(kronecker(level0[j][1],5)==-1, for(k=1, length(Quotring[j]),\
    res0=Quotring[j][k]; if(gcd(norm(res0), level0[j][1])==1,,\
          res=concat(res, [[res0, 1]]))),\
                for(k=1, length(Quotring[j]), res0=Quotring[j][k];\
        if(gcd(res0, level0[j][1])==1,, res=concat(res, [[res0, 1]]))));\
                      Proj0[j]=res); print(Proj0);

Proj=[];\
if(level1==1, Proj=Proj0[1], res0=vector(length(Proj0[1]));\
for(j=1, length(Proj0[1]), res0[j]=[Proj0[1][j]]); res0; j=2;\
while(j<=level1, Proj=[]; for(i=1, length(res0), for(k=1, length(Proj0[j]),\
res=concat(res0[i], [Proj0[j][k]]); Proj=concat(Proj, [res])));\
res0=Proj; j=j+1)); print(Proj);

\\This routine embeds the quaternion q into the ring M_2(O/level).
\\Return 1 if x^2+1 has a root mod level and the first solution.
alpha=vector(level1);\
for(j=1, level1, res=[0];\
 k=1; while(k<=length(Quotring[j])&&\
      !((Quotring[j][k]^2+1)%(level0[j][1]^level0[j][2])==0), k=k+1);\
  if(k>length(Quotring[j]),, res=concat(1, Quotring[j][k]));\
   alpha[j]=res); alpha;

\\When x^2+1 has no root mod level, this routine returns a matrix
\\J=[a, b; b, -a] such that J^2=-[1, 0; 0, 1] mod level;
J=vector(level1);\
for(j=1,level1, res=res0=[0, 0; 0, 0];\
if(level0[j][1]==2, J[j]=matredmod([1, 2 + 9*w; 2 + 10*w, -1], j);\
J0=matredmod([27 + 23*w, 31 + 4*w; 4*w, 6 + 9*w], j),\

if(alpha[j][1]==1, , i=2; k=1;\
res=[Quotring[j][i], Quotring[j][k]; Quotring[j][k], -Quotring[j][i]];\
until(!(matredmod(res-[0, -1; 1, 0], j)==[0, 0; 0, 0]) &&\
matredmod(res^2+[1, 0; 0, 1], j)==[0, 0; 0, 0], k=k+1;\
if(k>length(Quotring[j]), i=i+1; k=1);\
res=[Quotring[j][i], Quotring[j][k]; Quotring[j][k], -Quotring[j][i]]);\
J[j]=res))); J; J0; print(J":"J0);

\\This routine embeds the quaternion q into the ring M_2(O/level).
P=[1, 0, 1/2, 0; 0, 0, 1/2, 1/2; 0, 1, 1/2, w/2; 0, 0, 1/2, conj(w)/2];
embed(q, j, res)=res=0;\
if(level0[j][1]==2, x0=matsolve(P, q~);\
                       res=matredmod(x0[1]*[1, 0; 0, 1]+x0[2]*J[j]+\
                         x0[3]*J0+x0[4]*[0, -1; 1, 0], j),\
   q0=vectredmod(q, j); if(alpha[j][1]==1,\
       res=matredmod([q0[1]+ alpha[j][2]*q0[2], -q0[3]-alpha[j][2]*q0[4];\
               q0[3]-alpha[j][2]*q0[4], q0[1]-alpha[j][2]*q0[2]], j),\
       res=matredmod(q0[1]*[1, 0; 0, 1]+q0[2]*[0, -1; 1, 0]+q0[3]*J[j]+\
               q0[4]*([0, -1; 1, 0]*J[j]), j))); res;

\\Defining the action of a quaternion on an element of P_1(O/level).
act0(q, c0, j, res3)=res3=((embed(q, j)*c0~)~)%(level0[j][1]^level0[j][2]);\
    if(res3==[0,0],, if(gcd(norm(res3[1]), level0[j][1])==1,\
            res3=(res3/res3[1])%(level0[j][1]^level0[j][2]),\
            res3=(res3/res3[2])%(level0[j][1]^level0[j][2]))); res3;
act(q, c, res5)=res5=0;\
  if(level1==1, res5=act0(q, c, 1), res5=vector(level1);\
      for(j=1, level1, res5[j]=act0(q, c[j], j))); res5;

\\This action is used to find the degenerated hecke operator at the
\\prime $p$ dividing the level.

act2(q, c, j, res4)=res4=((embed(q, j)*c~)~)%level0[j][1]; res4;
\\This routine returns the operator $U(p)$ at a prime $p$ dividing
\\the level.

print(act(icosiangroup[1], Proj[1]));

\\This returns the orbit of a point in $P^1(O_K/level)$.
orbit(x,t)=t=[];\
 for(k=1, 60, t=concat(t, [act(icosiangroup[k], x)]));\
      t=eval(Set(t)); t;
print(orbit(Proj[1]));

\\Determining the global array orbits.
orbits=[orbit(Proj[1])];\
  tt=eval(setminus(Set(Proj), Set(orbits[1])));\
    until(length(tt)==0, orbits=concat(orbits, [orbit(tt[1])]);\
        tt=eval(setminus(Set(tt), Set(orbit(tt[1]))))); print(orbits);

\\Here is a fundamental domain of the action of the icosians on
\\$P^1(O_K/level)$.
FD=[];\
for(j=1, length(orbits), FD=concat(FD, [orbits[j][1]])); print(FD);

\\Computing the action of tp.
\\We first determine the number of operators whose norm does not divide the
\\level.

Hecke=[];\
for(p=1, length(tp), heckeop=tp[p];\
       res=nfeltreduce(field, fieldembed(hnorm(tp[p][1])),\
                idealhnf(field, level));\
  if(gcd(res, idealnorm(field, level))==1,,l0=1;\
     while(gcd(res, level0[l0][1])==1, l0=l0+1)); l0; print(p":"l0);\
Tp=matrix(length(FD), length(FD));\
for(k1=1, length(FD), for(k2=1, length(FD), counter=0;\
  if(level1==1, for(k3=1, length(heckeop),\
     if(act2(heckeop[k3],FD[k1],l0)==[0,0],,\
        if(!(setsearch(Set(orbits[k2]), act(heckeop[k3],FD[k1]))==0),\
             counter=counter+1))),\
               for(k3=1, length(heckeop),\
    if(act2(heckeop[k3],FD[k1][l0],l0)==[0,0],,\
        if(!(setsearch(Set(orbits[k2]), act(heckeop[k3],FD[k1]))==0),\
             counter=counter+1)))); Tp[k1,k2]=counter)); print(Tp);\
                  Hecke=concat(Hecke, [Tp])); Hecke;

vectoalg0(u)=res; if(length(u)==1, res=u[1], res=u[1]+u[2]*w0); res;
\\This function, applied to a polynomial, returns either an irreducible factor
\\of degree 3 or the discriminant vector of all the quadratic factors.

irredfact(pol, res)=res=0;\
  polfact=factor(pol); h=1; discvector0=[];\
    while(h<=matsize(polfact)[1] && poldegree(polfact[h, 1])<=2,\
      if(poldegree(polfact[h, 1])==2, D=quaddisc(poldisc(polfact[h, 1]));\
         discvector0=concat(discvector0, D), ); h=h+1); h; discvector0;\
    if(h>matsize(polfact)[1], res=[0, discvector0], res=[1, polfact[h, 1]]);\
                                         res;

s=1; discvector=[]; factofdeg3=[];\
while(s<=length(Hecke) && irredfact(charpoly(Hecke[s], x))[1]==0,\
          discvector=concat(discvector, irredfact(charpoly(Hecke[s], x))[2]);\

                 s=s+1); s; discvector;\
 if(s>length(Hecke), discvector=eval(Set(discvector)),\
 for(k=1, length(Hecke),\
factofdeg3=concat(factofdeg3, [factor(charpoly(Hecke[k], x))])); factofdeg3);\
                 discvector; factofdeg3;
heigen=[];\
if(!(factofdeg3==[]), heigen=factofdeg3,\
    if(length(discvector)>=2, heigen=discvector,\
     if(length(discvector)==0, splt=1; spltfield=nfinit(v-1), splt=0;\
      w0=quadgen(discvector[1]); spltfield=nfinit(v^2-trace(w0)*v+norm(w0)));\
rts=nfroots(spltfield, charpoly(Hecke[1], x)); print(rts);\
res0=[vectoalg0(nfalgtobasis(spltfield, rts[1]))];\
for(i=2, length(rts),\
res0=concat(res0, vectoalg0(nfalgtobasis(spltfield, rts[i])))); res0;\
print(res0);\
P=matker(Hecke[1]-res0[1]*matid(matsize(Hecke[1])[1]));\
for(i=2, length(res0),\
P=concat(P, matker(Hecke[1]-res0[i]*matid(matsize(Hecke[1])[1])))); P;\
j=2; while(j<=length(Hecke), B=P^-1*Hecke[j]*P;\
if(matisdiagonal(B), Q=matid(matsize(Hecke[j])[1]), print(B);\
rts=nfroots(spltfield, charpoly(Hecke[j], x)); print(rts);\
res1=[vectoalg0(nfalgtobasis(spltfield, rts[1]))];\
for(i=2, length(rts),\
res1=concat(res1, vectoalg0(nfalgtobasis(spltfield, rts[i])))); res1;\
print(res1);\
Q=matker(B-res1[1]*matid(matsize(Hecke[j])[1]));\
for(i=2, length(res1),\
Q=concat(Q, matker(B-res1[i]*matid(matsize(Hecke[j])[1])))); Q);\
P=P*Q; j=j+1); P; print(P);\
heckediag=vector(length(Hecke));\
for(j=1, length(Hecke), heckediag[j]=P^-1*Hecke[j]*P); heckediag;\
test=[]; for(j=1, length(heckediag),\

test=concat(test, matisdiagonal(heckediag[j]))); test; print(test);\
lambda=vector(matsize(Hecke[1])[1], i1, []);\
for(j=1, matsize(Hecke[1])[1], for(i=1, length(Hecke),\
       lambda[j]=concat(lambda[j], heckediag[i][j, j]))); lambda;\
       heigen=concat(heigen, lambda))); print(heigen);

\\WE USE THIS ROUTINE TO COMPUTE ALL THE QUATERNIONS NEEDED TO
\\COMPUTE A GIVEN HECKE OPERATOR.
\\Initial values
w=quadgen(5);
field=nfinit(z^2-z-1);
fieldembed(x)=Mod(real(x)+imag(x)*z, z^2-z-1);

\\Working with quaternions; we represent quaternions by
\\vectors [x,y,w,z].
hnorm(q)=q[1]^2+q[2]^2+q[3]^2+q[4]^2;
hconj(q)=[q[1],-q[2],-q[3],-q[4]];
hmult(a,b)=[a[1]*b[1]-a[2]*b[2]-a[3]*b[3]-a[4]*b[4],\
           a[1]*b[2]+a[2]*b[1]+a[3]*b[4]-a[4]*b[3],\
           a[1]*b[3]+a[3]*b[1]-a[2]*b[4]+a[4]*b[2],\
           a[1]*b[4]+a[4]*b[1]+a[2]*b[3]-a[3]*b[2] ];
hinv(q)=hconj(q)/hnorm(q)

\\This function returns the primes in $O_K$ lying above any prime
\\$p$ in $\Z$.
gen(pr, res)=res=0;\
   if(kronecker(pr,5)==-1,res=pr,N0=floor(sqrt(pr-1));\
            a=N0;res1=r^2-a*r+pr-a^2;\
       while(polisirreducible(res1),a=a-1;res1=r^2-a*r+pr-a^2);\
            res=floor(real(polroots(res1)))~;\
     if(res[1]==res[2], res=a+res[1]*w, res=[a,a]+w*res)); res;

\\This function reduces any matrix modulo pr, where pr is is
\\modprinit format.
matredmodpr(m,pr, res)=res=matrix(matsize(m)[1], matsize(m)[2]);\
for(i=1, matsize(m)[1], for(j=1, matsize(m)[2],\
   res0=nfeltreducemodpr(field, fieldembed(m[i,j]), pr);\
     res0=lift(res0); res[i,j]=res0[1]+res0[2]*w)); res;
vectredmodpr(v,pr,res1)=res1=vector(length(v));\
for(l=1, length(v), res0=nfeltreducemodpr(field,\
                           fieldembed(v[l]), pr);\
         res0=lift(res0); res1[l]=res0[1]+res0[2]*w); res1;

\\Every integral quaternion in this order is then given in the form:
quat(a,b)=a+w*b;
\\We get its norm in the form $Q+R*w$, where $Q$ and $R$ are the following
\\quadratic forms.
Q(a,b)= real(hnorm(a+b*w));
R(a,b)= imag(hnorm(a+b*w));

\\This function returns the norm of the ideals lying above a prime in $O_K$.
norm0(p)=if((kronecker(p,5)==1||kronecker(p,5)==0), p, p^2);

Primes=[];\
prs=primes(1440); for(i=1, length(prs), p=prs[i];\
            if(norm0(p)<=12000), Primes=concat(Primes, p))); Primes;

heckeops=[];\
 for(n=1, length(Primes), p=Primes[n]; read(comp1);\
         heckeops=concat(heckeops, [hecke])); heckeops;
write(hecke, heckeops);

\\This is the subroutine COMP1.GP we call in our main program which
\\compute the hecke operators.

Pr=if((kronecker(p,5)==-1||kronecker(p,5)==0), gen(p), gen(p)[1]);\
print(Pr); Pr0=nfmodprinit(field,idealfactor(field, fieldembed(Pr))[1,1]);
Pr1=if(norm(Pr)==Pr^2, Pr, norm(Pr)); print(Pr1);
\\This return a root of x^2+1 has a root mod level and the first solution.
alpha=res=0;\
 if((norm(Pr)-1)%4==0, G=idealstar(field, Pr0, flag=0);\
   res=nfeltpowmodpr(field, G[3][1],G[2][1]/4, Pr0);\
     res=lift(res); alpha=res[1]+res[2]*w, alpha=0); alpha;

\\When x^2+1 has no root mod level, this routine returns a matrix
\\J=[a, b; b, -a] such that J^2=-[1, 0; 0, 1] mod level;
J=res=res0=[0, 0; 0, 0];\
if(Pr==2, J=[matredmodpr([1, 2 + 9*w; 2 + 10*w, -1], Pr0),\
 matredmodpr([27 + 23*w, 31 + 4*w; 4*w, 6 + 9*w], Pr0)],\
if(alpha==0, i=2; k=1; res=[i, k; k, -i];\
until(!(matredmodpr(res-[0, -1; 1, 0], Pr0)==[0, 0; 0, 0]) &&\
matredmodpr(res^2+[1, 0; 0, 1], Pr0)==[0, 0; 0, 0], k=k+1;\
if(k>=norm(Pr), i=i+1; k=1); res=[i, k; k, -i]); J=res)); J; print(J);

\\This routine embeds a quaternion into $M_2(O_K/pr)$.

embed(q, res2)=res2=0;\
if(Pr==2, x0=matsolve(P, q~);\
                       res2=matredmodpr(x0[1]*[1, 0; 0, 1]+x0[2]*J[1]+\
                         x0[3]*J[2]+x0[4]*[0, -1; 1, 0], Pr0),\
   q0=vectredmodpr(q, Pr0); if(alpha==0,\
       res2=matredmodpr(q0[1]*[1, 0; 0, 1]+q0[2]*[0, -1; 1, 0]+q0[3]*J+\
               q0[4]*([0, -1; 1, 0]*J), Pr0),\
       res2=matredmodpr([q0[1]+ alpha*q0[2], -q0[3]-alpha*q0[4];\
               q0[3]-alpha*q0[4], q0[1]-alpha*q0[2]], Pr0))); res2;

\\This function returns of the equivalence class of $x$ in $P^1(O_K/Pr)$
\\in the form $[0,1]$ or $[1,a]$, with $a\in F_q$.
eqcls(u,res3)=res3=0;\
 if(u[1]==0, res3=[0,1], res3=[1, u[2]/u[1]]%Pr1); res3;

N=real(Pr); N1=floor(sqrt(N));

hecke=[];\
S=[];\
for(x1=0,N1, N2=floor(sqrt(N-x1^2));\
  for(x2=0, N2, N3=floor(sqrt(N-x1^2-x2^2));\
    for(x3=-N3, N3, N4=floor(sqrt(N-x1^2-x2^2-x3^2));\
      for(x4=-N4, N4, N5=floor(sqrt(N-x1^2-x2^2-x3^2-x4^2));\
        for(y1=-N5, N5, N6=floor(sqrt(N-x1^2-x2^2-x3^2-x4^2-y1^2));\
          for(y2=-N6, N6, N7=floor(sqrt(N-x1^2-x2^2-x3^2-x4^2-y1^2-y2^2));\
             for(y3=-N7, N7, x=[x1,x2,x3,x4]; y=[y1,y2,y3,r];\
                res0=Q(x,y)-real(Pr); if(polisirreducible(res0)==0,\
                   res=floor(real(polroots(res0)));\
                for(i=1,2, y0=[y1,y2,y3,res[i]]; res4=embed(quat(x,y0))[1,];\
             if((R(x,y0)==imag(Pr))&&(!(res4==[0,0])), res4=eqcls(res4);\
          if(setsearch(S, res4)==0, S=setunion(S, Set([res4]));\
        hecke=concat(hecke, [quat(x,y0)]);\
     print(length(hecke)":"norm(Pr)-length(S)+1);\
 if(length(hecke)==norm(Pr)+1, return(hecke)))))))))))));

94 1. COMPUTING HILBERT MODULAR FORMS.

print(hecke);

CHAPTER 2 Some numerical results on the Oda conjecture.

Let f be a newform of weight 2 and level 𝔠 on GL_{2}(𝔸_{ℚ}) with coefficients in ℚ. The construction of Eichler–Shimura associates to f an elliptic curve E_{f}/ℚ whose complex points are given by

E_{f}(ℂ) = ℂ/Λ_{f},

where Λ_{f}, the period lattice of f, is canonically isomorphic to H_{1}(E_{f}, ℤ) (see Knapp [34, Chap. XI]). The elliptic curve E_{f} is a quotient of the Jacobian variety of the modular curve X_{0}(𝔠). The Taniyama–Shimura– Weil conjecture over ℚ, which is now a theorem thanks to the work of Wiles [57] and several people, can be stated as follows. TSW–conjecture: For any elliptic curve E over ℚ, there is a newform f of weight 2 and level 𝔠_{E} on GL_{2}(𝔸_{ℚ}) and an isogeny

ϕ : E_{f} → E.

(We refer to Diamond and Im [20, Theorem 13.0.5.] for several equivalent statements.)

Let F be a totally real number field of degree g > 1. In an attempt to generalize the Eichler–Shimura construction to Hilbert modular forms over F, Oda has proposed a conjecture that circumvents the lack of a good substitute for the Jacobian variety in higher dimension. This proceeds as follows. Let 𝔠 be an integral ideal in F and X_{0}(𝔠) the modular variety of level 𝔠. By Deligne [16, 17], the middle dimensional

96 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

cohomology H^{g}(X_{0}(𝔠), ℚ) carries a mixed Hodge structure (see the text for the precise definitions). Let (W_{k})_{k∈ℤ} be its weight filtration, and consider W_{g}H^{g}(X_{0}(𝔠), ℚ). This has a homogeneous Hodge structure of weight g on which the Hecke operators act in a natural way. Let f be a newform of weight 2 and level 𝔠. We let K_{f} be the number field generated by the Fourier coefficients of f. Then we have the natural homomorphism

𝕋 ⟶ K_{f}
T_{𝔭} ⟼ a_{𝔭},

where 𝕋 is the (Hecke) ℚ-algebra generated by all the operators T_{𝔭} and S_{𝔭}. Put

H^{g}(M_{f}, ℚ) = W_{g}H^{g}(X_{0}(𝔠), ℚ) ⊗_{𝕋} K_{f}.

This is a rational homogeneous Hodge structure of weight g with a homomorphism

θ_{f} : K_{f} ↪ End(H^{g}(M_{f}, ℚ)).

Oda conjecture: For each newform f of weight 2 and level 𝔠, there exists an abelian variety A_{f} of dimension [K_{f} : ℚ] defined over F, with good reduction outside 𝔠, and a homomorphism

θ_{f} : K_{f} ↪ End(A_{f}) ⊗ ℚ

such that there is an isomorphism of K_{f}–Hodge structures

⨂_{σ∈Gal(F/ℚ)}H^{1}(A_{f}^{σ}(ℂ), ℚ) → H^{g}(M_{f}, ℚ).

When F is a quadratic field, Oda [40, Chap. II. sec.7] has proved that the Hodge structure H^{2}(M_{f}, ℚ) is the tensor product of two rational Hodge structures of weight 1. However the field of definition of the corresponding abelian variety is unknown. In this section, we try to provide some numerical evidence for this conjecture by making use of

1. A BRIEF REVIEW OF HODGE STRUCTURES. 97

the computations that we have done in the first chapter. Namely, for F = ℚ(\sqrt{5}), we give two examples which show that the period moduli of Oda agree to a certain precision with the period moduli of some specific elliptic curves that are known to be defined over F = ℚ(\sqrt{5}). In order to do so, we need to recall the main results of Oda on his own conjecture. This requires a brief review on Hodge structures.

1. A brief review of Hodge structures.

References: Deligne [16, 17].

Let \underline{S} = Res_{ℂ/ℝ}𝔾_{m} be the algebraic group obtained by restriction of scalars. We denote by w : 𝔾_{m/ℝ} → \underline{S} the morphism which corresponds to the natural inclusion ℝ^{×} ↪ ℂ^{×}, and by t : \underline{S} → G_{m/ℝ} the inverse of the norm morphism. We have t ∘ w(x) = x^{−2}.

Let H_{ℝ} be a finite dimensional real vector space, and H = H_{ℂ} = H_{ℝ} ⊗ ℂ its complexification. We let E be a subset of ℤ × ℤ.

Definition 2.1. A real Hodge structure on H_{ℝ} is a bigraduation

H = ⨁_{p, q} H^{p, q}, such that \overline{H^{q, p}} = H^{p, q}.

We say that the Hodge structure {H_{ℝ}, H^{p, q}}

a) is of type E if H^{p, q} = 0 for (p, q) ∉ E;

b) has weight n if H^{p, q} = 0, for p + q ≠ n. We call h^{p, q} = dim H^{p, q} the Hodge numbers of {H_{ℝ}, H^{p, q}}.

By making use of the theory of multiplicative groups, Definition 2.1 is equivalent to the following one.

98 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

Definition 2.2. Let H_{ℝ} be a finite dimensional real vector space. A real Hodge structure is a representation of real algebraic groups

h : \underline{S} → GL(H_{ℝ}).

We say that h has weight n if h ∘ w(x) = x^{n}. We call C = h(i) the Weil operator.

Now, let H_{ℤ} be a free ℤ-module of finite rank, and H = H_{ℂ} = H_{ℤ} ⊗ ℂ its complexification. For every pair of integers (p, q), pick a complex linear subspace H^{p, q} ⊂ H.

Definition 2.3. We say that the data {H_{ℤ}, H^{p, q}} is a Hodge structure if {H_{ℝ} = H_{ℤ} ⊗ ℝ, H^{p, q}} is a real Hodge structure. Similarly, a ℚrational Hodge structure consists of the data of a ℚ-vector space H_{ℚ} together with a bigraduation H^{p, q}, (p, q) ∈ ℤ × ℤ of H_{ℚ} ⊗ ℂ such that {H_{ℚ} ⊗ ℝ, H^{p, q}} is a real Hodge structure.

Example 2.4. 1) The simplest example of Hodge structure, coming from algebraic geometry, is provided by the Tate Hodge structure ℤ(n) of type (−n, −n) given by

ℤ(n)_{ℂ} = ℂ, and ℤ(n)_{ℤ} = (2πi)^{n}ℤ ⊂ ℂ.

2) Let X be a Kähler manifold. We let

H_{ℤ} = H^{n}(X, ℤ)/(torsion), H_{ℂ} = H_{ℤ} ⊗ ℂ ≅ H^{n}(X, ℂ),

and H^{p, q} be the cohomology of type (p, q) in the Hodge decomposition of H^{n}(X, ℂ). Then {H_{ℤ}, H^{p, q}} is a Hodge structure of weight n. (See [27, Chap.0, sec.7]).

3) Let A be a complex abelian variety, and A(ℂ) its complex points. Since A is projective, A(ℂ) is a Kähler manifold; so the cohomology group H_{ℤ} = H^{1}(A, ℤ) carries a Hodge structure of weight 1. This is a polarized Hodge structure, and the assignment A ↦ H^{1}(A, ℤ) is an equivalence of categories (see Section 4 for more details).

1. A BRIEF REVIEW OF HODGE STRUCTURES. 99

From Definition 2.2, one gets a natural way to define the notion of morphisms and tensor products of Hodge structures. Indeed, a morphism of Hodge structures will correspond to the morphism of the underlying representations; likewise for tensor product. The category of ℚ–Hodge structures is abelian.

Definition 2.5. A polarization of a Hodge structure of weight n is a morphism of Hodge structures ψ : H_{ℤ} ⊗ H_{ℤ} → ℤ(−n) such that, on H_{ℝ} = H_{ℤ} ⊗ ℝ, the real bilinear form (x, y) ↦ (2πi)^{n}ψ(x, Cy) is symmetric positive definite. Note that ψ is then symmetric or antisymmetric according to the parity of the weight n.

In terms of Definition 2.2, a polarization of a real Hodge structure H_{ℝ} of weight n is a bilinear form ψ on H_{ℝ}, invariant by the compact subgroup ker(t) of \underline{S}, such that the form (x, y) ↦ ψ(x, Cy) is symmetric positive definite. So, a polarization of a Hodge structure H_{ℤ} of weight n is a polarization ψ of the real Hodge structure H_{ℝ} = H_{ℤ} ⊗ ℝ which is integral valued on H_{ℤ}.

Example 2.6. Let A be a complex abelian variety. The intersection form gives a polarization of the homogeneous Hodge structure H^{1}(A, ℤ). Let ψ be polarization of H^{1}(A, ℤ). It is a perfect pairing; therefore it induces a ℤ–linear injection of H^{1}(A, ℤ) into its ℤ–dual. As a result, we get a polarization of A.

Remark 2.7. In general, for any Noetherian subring A of ℝ, one defines an A-Hodge structure of weight n to be a finitely generated Amodule H_{A} such that H_{ℝ} = H_{A} ⊗_{A} ℝ has a real Hodge structure of weight n. A polarization of an A–Hodge structure H_{A} is a polarization of the underlying real Hodge structure which is A–valued on H_{A}.

100 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

Now, let G be a reductive group over ℚ, and t and w two ℚmorphisms

𝔾_{m} ⟶^{w} G ⟶^{t} 𝔾_{m}

with t ∘ w(x) = x^{−2} and w central. Furthermore, assume we have a commutative diagram

𝔾_{m/ℝ} ⟶^{w} \underline{S} ⟶^{t} 𝔾_{m/ℝ}
∥ ↓^{h} ∥
𝔾_{m/ℝ} ⟶^{w} G_{ℝ} ⟶^{t} 𝔾_{m/ℝ}.

Let ρ : G → GL(V_{ℚ}) be a representation defined over ℚ. We say that ρ is homogeneous of weight n if ρ∘w(x) = x^{n} ·1_{V}. Every representation is a sum of homogeneous representations.

We can let G act on ℚ(n)_{ℚ} by the rule g ∗ x = t(g)^{n} · x. This definition is made compatible with the commutative square above as \underline{S} acts on ℚ(n)_{ℝ} in a similar way. Definition 2.8. A polarization of ρ : G → GL(V_{ℚ}) is a G-invariant form

(2πi)^{n}ψ : V_{ℚ} ⊗ V_{ℚ} → ℚ(n)

such that (x, ρ ∘ h(y)) is symmetic positive definite on V_{ℝ}.

Proposition 2.11 in Deligne [17] gives necessary and sufficient conditions for a representation to be polarizable. We see that, if V_{ℚ} has a rational Hodge structure, any polarization of a representation V_{ℚ} induces a polarization of this rational Hodge structure on V_{ℚ}. Definition 2.9. A mixed Hodge structure is a triple (H_{ℤ}, W, F) which consists of:

(i) A finitely generated ℤ-module H_{ℤ}. (ii) A finite increasing filtration W_{n} of H_{ℚ} = H_{ℤ}⊗_{ℤ}ℚ called the weight

2. COHOMOLOGY OF HILBERT MODULAR SURFACES. 101

filtration. (iii) A finite decreasing filtration F^{p} of H_{ℂ} = H_{ℤ}⊗_{ℤ}ℂ called the Hodge filtration. We require that the filtration induced on Gr_{n}^{W} H_{ℂ} by F provides Gr_{n}^{W} H_{ℚ} with a homogeneous Hodge structure of weight n.

In [17], Deligne proved that the category of mixed Hodge structures is abelian and that the cohomology of any finite scheme over ℂ has a mixed Hodge structure and that this construction is functorial. The main example of a mixed Hodge structure we will be dealing with is the subject of the next section.

2. Cohomology of Hilbert modular surfaces.

Reference Oda [40, Chap. I.] and [41].

We assume in this section that the modular surface X_{0}(𝔠) is connected but this restriction could be removed. Recall that we have intersection pairing

⟨ , ⟩ : H_{2}(X_{0}(𝔠)) × H_{2}(X_{0}(𝔠)) → ℚ

which is non-degenerate. By making use of Deligne [18, sec. 8], Oda computed the weight filtration of the Hodge structure of H^{2}(X_{0}(𝔠), ℚ):

W_{k}H^{2}(X′_{0}(𝔠), ℚ) = 0, for k ≤ 1,
W_{2}H^{2}(X′_{0}(𝔠), ℚ) = W_{3}H^{2}(X′_{0}(𝔠), ℚ) = j^{∗}H^{2}(X_{0}(𝔠), ℚ),

where j is the natural inclusion. We put

ℍ^{2}(X_{0}(𝔠), ℚ) = j^{∗}H^{2}(X_{0}(𝔠), ℚ).

102 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

One gets the Hodge filtration on ℍ^{2}(X_{0}(𝔠), ℚ) by using its description in terms of Hilbert modular forms. To this end, let ε_{1} (resp. ε_{2}) be the involution on X_{0}(𝔠) induced by the maps ((z_{1}, z_{2}) ↦ (ε̄z̄_{1}, εz_{2})) and ((z_{1}, z_{2}) ↦ (εz_{1}, ε̄z̄_{2})), where ε is a unit in O_{F} such that ε > 0 and ε̄ < 0. Let f be in S_{2}(𝔠). We can define a holomorphic 2-form on X_{0}(𝔠), by putting

ω_{f} = (2πi)^{2}f(z_{1}, z_{2})dz_{1} ∧ dz_{2} on Γ\H^{2}.

This gives a linear map S_{2}(𝔠) → ℍ^{2}(X_{0}(𝔠), ℂ). We define

η_{f} = ε_{1}^{∗}ω_{f},
η′_{f} = ε_{2}^{∗}ω_{f}.

Oda obtained the Hodge filtration of ℍ^{2}(X_{0}(𝔠), ℚ).

Theorem 2.10. [40, Chap.I, sec.1] ℍ^{2}(X_{0}(𝔠), ℂ) is the direct sum of (i) its (2, 0)-component {ω_{f} : f ∈ S_{2}(𝔠)} (ii) its (0, 2)-component {ω̄_{f} : f ∈ S_{2}(𝔠)} (iii) its (1, 1)-component {η_{f} + η′_{g} : f, g ∈ S_{2}(𝔠)} ⊕ W, where W is the 2-dimensional space span by the forms

w_{1} = −\frac{1}{2π} \frac{dx_{1} ∧ dy_{1}}{y_{1}^{2}} and w_{2} = −\frac{1}{2π} \frac{dx_{2} ∧ dy_{2}}{y_{2}^{2}}

.

We are only interested in the orthogonal complement of W which arises from cusp forms. We call it the cuspidal cohomology and denote it by ℍ_{sp}^{2}(X_{0}(𝔠), ℚ). We denote by ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} the part of ℍ_{sp}^{2}(X_{0}(𝔠), ℚ) that is described by newforms and call it the new part of ℍ_{sp}^{2}(X_{0}(𝔠), ℚ). ℍ_{sp}^{2}(X_{0}(𝔠), ℚ) and ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} are both equipped with some homogeneous Hodge structures polarized by the restriction of the bilinear form coming from Poincaré duality.

3. THE HODGE STRUCTURE ATTACHED A HILBERT NEWFORM. 103

Remark 2.11. Our definition of the new part of the cuspidal cohomology is equivalent to the one in Oda [41], by Proposition 3.3 in [41].

3. The Hodge structure attached a Hilbert newform.

In the previous section, we obtained a homogeneous Hodge structure together with a bilinear pairing

Ψ : ℍ_{sp}^{2}(X_{Γ}, ℚ)^{new} × ℍ_{sp}^{2}(X_{Γ}, ℚ)^{new} → ℚ

coming from Poincaré duality. We will now define an action of the Hecke algebra on ℍ_{sp}^{2}(X_{Γ}, ℚ)^{new}. This action is compatible with the polarization Ψ. As a consequence, we will get that ℍ_{sp}^{2}(X_{Γ}, ℚ)^{new} decomposes as a direct sum of polarized Hodge structures.

The action of the Hecke algebra is given as usual via algebraic correspondences. Let 𝔭 be a prime ideal in F such that (𝔭, 𝔠) = 1. Then, as in the one dimensional case, the diagram

\xymatrix{ & X′_{0}(𝔭𝔠) \ar[dl] \ar[dr] \\ X′_{0}(𝔠) &  & X′_{0}(𝔠)}

with arrows corresponding to natural projections, gives rise to a correspondence on X′_{0}(𝔠) × X′_{0}(𝔠) which extends to a correspondence C on X_{0}(𝔠) × X_{0}(𝔠). Let p_{1}, p_{2} : C → X_{0}(𝔠) be the canonical projections. They are finite flat morphisms, étale outside a finite set of points. Therefore, we have an action of T_{𝔭} on H_{2}(X′_{0}(𝔠), ℚ) given by

T_{𝔭} · c = p_{2}(p_{1}^{−1}c), c ∈ H_{2}(X′_{0}(𝔠), ℚ),

and an action on H_{DR}^{2}(X′_{0}(𝔠), ℚ) given by

T_{𝔭} · ω = (p_{2})_{∗}(p_{1}^{∗}ω), ω ∈ H_{DR}^{2}(X′_{0}(𝔠), ℚ),

where p_{1}^{∗} and (p_{2})_{∗} are the usual pull back and trace maps respectively. To avoid problems with quotient singularities, we take a torsion free normal subgroup Γ′ of Γ and put

H_{DR}^{2}(X′_{0}(𝔠), ℚ) = H_{DR}^{2}(X_{Γ′}, ℚ)^{Γ/Γ′}.

This definition does not depend on Γ′ as one can check. To ensure self-adjointness, we restrict to primes 𝔭 not dividing the level 𝔠. Let T be the image of the Hecke ℚ–algebra in End(ℍ^{2}(X_{0}(𝔠), ℚ)). By symmetry of Hecke correspondences, we have

Ψ(T · ω_{1}, ω_{2}) = Ψ(ω_{1}, T · ω_{2}), ω_{1}, ω_{2} ∈ H_{DR}^{2}(X_{0}(𝔠), ℚ), T ∈ T.

Theorem 2.12. [40, Theorem 1.12] T acts on the homogenous polarized Hodge structure ℍ^{2}(X_{0}(𝔠), ℚ) of weight 2 as endomorphisms of Hodge structures. This action preserves ℍ_{sp}^{2}(X_{0}(𝔠), ℚ) and ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new}.

As a consequence of the Multiplicity one Theorem (see Miyake [39]) and the fact that S_{2}(𝔠)^{new} has a basis of eigenforms we have the following lemma.

Lemma 2.13. [41, Proposition 3.3.] T is a semi-simple ℚ-algebra.

Since T is a semi-simple ℚ-algebra, we can write it as a direct sum of number fields

T = ⨁_{i=1}^{n} K_{i}.

3. THE HODGE STRUCTURE ATTACHED A HILBERT NEWFORM. 105

Note that since each T ∈ T is self–adjoint with respect to the Peterson inner product, it has real eigenvalues, therefore each K_{i} is totally real. Let

ℍ_{sp}^{2}(X_{0}(𝔠), ℂ)^{new} = H^{2, 0} ⊕ H^{1, 1} ⊕ H^{0, 2}

be the Hodge decomposition of ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new}. By Theorems 2.10 and 2.12, each summand has the same isotypic components as S_{2}(𝔠)^{new} as a 𝕋-module. And, since S_{2}(𝔠)^{new} admits a basis of eigenforms, putting [K_{i} : ℚ] = d_{i}, we get ∑ d_{i} = dim S_{2}(𝔠)^{new}. We choose a set of primitive idempotents {e_{1}, …, e_{n}} such that K_{i} = e_{i}T = Te_{i} = e_{i}Te_{i}. Then,

ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} = ⨁_{i=1}^{n} e_{i}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new}.

Since the action of T on ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} preserves Hodge type, each e_{i}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} is a sub-Hodge structure. Moreover, by symmetry of Hecke correspondences, we see that the above decomposition is in fact a direct sum of polarized Hodge structures, the polarization of e_{i}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} being obtained by restriction of the form Ψ. We denote by

θ_{i} : K_{i} → End(e_{i}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new})

the action of K_{i} on e_{i}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new}. Take an idempotent e_{i} ∈ {e_{1}, …, e_{n}} and let f ∈ e_{i}S_{2}(𝔠)^{new} be a normalized eigenform. Then f defines an embedding σ : K_{i} → ℂ by t(f) = σ(t)f. Since K_{i} ⊗_{ℚ} ℂ = ⨁_{σ} ℂ with the summands corresponding to the embeddings σ : K_{i} → ℂ, we have

e_{i}S_{2}(𝔠)^{new} = ⨁_{σ} ℂf^{σ},

where the f^{σ} are the normalized eigenforms with t(f^{σ}) = σ(t)f^{σ}. We call the f^{σ} the companions of f. Now, we choose a set E = {f_{1}, …, f_{n}} of normalized eigenforms such that f_{i} ∈ e_{i}S_{2}(𝔠)^{new}. Each f_{i} then

106 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

corresponds to a unique field K_{i}. So, for a normalized eigenform f ∈ e_{i}S_{2}(𝔠)^{new}, we can put e_{i} = e_{f}, K_{i} = K_{f}, Ψ_{i} = Ψ_{f}, θ_{i} = θ_{f} and define

H^{2}(M_{f}, ℚ) := e_{f}ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new}.

This symbol M_{f} though very suggestive as it seems to refer to a certain conjectural motive associated to f, of which we only study the Hodge realization, will be used only in combination with H^{2}(M_{f}, ℚ).

Theorem 2.14. [41, Lemma 3.5.] Corresponding to T = ⨁_{f∈E} K_{f}, there is a direct sum decomposition

ℍ_{sp}^{2}(X_{0}(𝔠), ℚ)^{new} = ⨁_{f} H^{2}(M_{f}, ℚ)

of polarized Hodge structures. Moreover, if K_{f} acts on H^{2}(M_{f}, ℚ) via θ_{f}, then

Ψ_{f}(θ_{f}(t)x, y) = Ψ_{f}(x, θ_{f}(t)y)

with Ψ_{f} the polarization of H^{2}(M_{f}, ℚ).

The involutions ε_{1} and ε_{2} defined on X_{0}(𝔠) and Y_{0}(𝔠) induce an action of ℤ/2ℤ × ℤ/2ℤ on ℍ_{sp}^{2}(X_{0}(𝔠), ℚ) which commutes with the action of 𝕋. Therefore, we can split H^{2}(M_{f}, ℚ) as

H^{2}(M_{f}, ℚ) = ⨁_{s,s′∈{+, −}} H^{2}(M_{f}, ℚ)_{ss′},

where ε_{1}^{∗} (resp. ε_{2}^{∗}) acts on H^{2}(M_{f}, ℚ) as s · id (resp. s′ · id).

Proposition 2.15. [24, Proposition 6.3] For every newform f ∈ S_{2}(𝔠) we have rank_{K_{f}} H^{2}(M_{f}, ℚ)_{ss′} = 1. Furthermore, Ψ_{f} = Tr_{K_{f} /ℚ}ψ_{f} with

ψ_{f} : H^{2}(M_{f}, ℚ) × H^{2}(M_{f}, ℚ) → K_{f}

3. THE HODGE STRUCTURE ATTACHED A HILBERT NEWFORM. 107

which, with respect to a suitable basis e_{ss′} ∈ H^{2}(M_{f}, ℚ)_{ss′}, is of the form

(\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) .

We call such a basis (e_{ss′}) a canonical basis.

Define

ℍ_{2}^{sp}(X_{0}(𝔠), ℚ) = {γ ∈ Im(H_{2}(X′_{0}(𝔠), ℚ) → H_{2}(Z_{0}(𝔠), ℚ)) :
∫_{γ} w = 0, ∀w ∈ W } .

By duality, we have a direct sum

ℍ_{2}^{sp}(X_{0}(𝔠), ℚ)^{new} = ⨁_{f} H_{2}(M_{f}, ℚ),

and each H_{2}(M_{f}, ℚ) splits into four eigenspaces H_{2}(M_{f}, ℚ)_{ss′}.

Definition 2.16. Let (e_{ss′}) be a canonical basis of H^{2}(M_{f}, ℚ), (γ_{ss′}) a dual basis in H_{2}(M_{f}, ℚ) and g a companion of f. The periods of g are given by

Ω_{ss′}(g) = ∫_{γ_{ss′}} ω_{g}.

The following period relation which is due to Oda will be needed later.

Theorem 2.17 (Hodge–Riemann bilinear relation). [40, Theorem 4. 4.]:

Ω_{++}(g)Ω_{−−}(g) = −Ω_{+−}(g)Ω_{−+}(g) = \frac{1}{4} ∫_{X′_{0}(𝔠)} ω_{g} ∧ ω̄_{g}.

4. Hilbert-Blumenthal abelian varieties.

We start with a totally real number field K of degree d, and let O_{K} be its ring of integers. With this notation, we have in mind the field K_{f} generated by the eigenvalues a newform f.

Definition 2.18. A complex Hilbert-Blumenthal abelian variety of dimension d is a pair (A, θ), where:

(i) A is a complex abelian variety, (ii) θ : O_{K} ↪ End(A) defines an action of O_{K} on A.

The Lie algebra of any complex Hilbert-Blumenthal abelian variety satisfies the following lemma.

Lemma 2.19. Let A be a complex Hilbert-Blumenthal abelian variety of dimension d. Then, Lie(A/ℂ) is locally free O_{K} ⊗_{ℤ} ℂ-module of rank 1.

Let A be a complex abelian variety. From the general theory of complex abelian varieties, we know that Lie(A/ℂ) is the universal covering space of A(ℂ) so that

Lie(A/ℂ)/Λ ≅ A(ℂ),

where Λ is a lattice in Lie(A/ℂ) canonically isomorphic to H_{1}(A, ℤ). Moreover H_{1}(A, ℤ) admits a Riemann form which, by Poincaré duality, gives a polarization of the Hodge structure of type {(1, 0), (0, 1)} on H^{1}(A, ℤ). We have the following well known theorem.

Theorem 2.20 (Riemann). The map A ↦ H^{1}(A, ℤ) is an equivalence between the category of abelian varieties and the one of polarized Hodge structures of type {(0, 1), (1, 0)}.

4. HILBERT-BLUMENTHAL ABELIAN VARIETIES. 109

Now, let (A, θ) be a Hilbert-Blumenthal abelian variety. We let ψ be the polarization of A given by the Riemann form which defines the algebraic structure on A. Then, the Rosati involution defined by ψ acts trivially on O_{K}. So letting

ψ^{∗} : H^{1}(A, ℤ) × H^{1}(A, ℤ) → ℤ

be the bilinear pairing induced from ψ, we get that

ψ^{∗}(θ^{∗}(a)ω_{1}, ω_{2}) = ψ^{∗}(ω_{1}, θ^{∗}(a)ω_{2}).

To avoid certain technical problems, we assume that the pairing ψ^{∗} is obtained as the composition of the trace map with some O_{K}–valued pairing.

Theorem 2.21. The map (A, θ) ↦ (H^{1}(A, ℤ), θ^{∗}) is an equivalence between the category of Hilbert-Blumenthal abelian varities with RM by O_{K} and the one of polarized Hodge structures of type {(0, 1), (1, 0)} with O_{K}-action.

We would like to determine the periods of the complex Hilbert-Blumenthal abelian variety (A, θ). To this end, let ω be a nowhere vanishing invariant differential form on A. Then, under the action of O_{K}, we have the decomposition

\underline{ω}_{A/ℂ} = H^{0}(A, Ω_{A/ℂ}^{1}) = ⨁_{σ} ω^{σ},

where σ runs over all the complex embeddings of K. As ℂ-vector spaces, we know that Lie(A/ℂ) and \underline{ω}_{A/ℂ} have dimension d, and they are duals of each other. Therefore, the differential form ω defines an O_{K} ⊗_{ℤ} ℂ isomorphism:

ω : Lie(A/ℂ) ≅ K ⊗ ℂ.

110 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

The period lattice L of A is then given by the image of H_{1}(A, ℤ) under that isomorphism. More explicitly, let ω_{1}, ω_{2} be an O_{K}-basis of H^{1}(A, ℤ) such that ψ^{∗}(ω_{1}, ω_{2}) = 1, and γ_{1}, γ_{2} its dual basis in H_{1}(A, ℤ). For each embedding σ : K ↪ ℂ, and i = 1, 2, put

Ω_{i}^{σ} = ∫_{γ_{i}} ω^{σ}.

Then, by Riemann period relations, the point τ = (τ_{σ})_{σ} ∈ (ℂ − ℝ)^{d}, given by

τ_{σ} = Ω_{2}^{σ}/Ω_{1}^{σ},

belongs to H^{d}. It determines the lattice L. The point τ ∈ H^{d} we just constructed does not depend on the choice of ω, however, it depends on the one of ω_{1} and ω_{2}. Indeed, let ω′_{1}, ω′_{2} be another O_{K}-basis of H^{1}(A, ℤ) such that ψ^{∗}(ω′_{1}, ω′_{2}) = 1. Then there exists γ ∈ SL_{2}(O_{K}) such that

(\begin{matrix}ω′_{1} \\ ω′_{2} \end{matrix}) = γ (\begin{matrix}ω_{1} \\ ω_{2} \end{matrix}) .

The resulting point which corresponds to (ω′_{1}, ω′_{2}) is then γ · τ. Therefore, SL_{2}(O_{K})\H^{d} parametrizes the period moduli of all Hilbert-Blumen-thal abelian varieties with O_{K}-action. In fact, the compactification of SL_{2}(O_{K})\H^{d} is the analytic space of the moduli stack M_{K} which parametrizes all semiabelian Hilbert-Blumenthal varieties with real multiplication by O_{K}. If we start instead with the cohomology H_{1}(A, ℚ), the abelian variety we get is then defined only up to ℚ-isogeny. So the above discussion then needs some modifications (see Oda [40, Chap. II.]).

Remark 2.22. Note that the variety M_{K} is pretty much distinct from the Hilbert modular variety X_{0}(𝔠). Indeed, each point on M_{K} parametrizes an isomorphism class of Hilbert–Blumenthal abelian varieties

5. THE ABELIAN VARIETY ASSOCIATED TO A HILBERT NEWFORM. 111

with multiplication by O_{K}. So when K = K_{f}, the field generated by the Fourier coefficients of a newform f, the piece H^{g}(M_{f}, ℚ) will correspond to some point on M_{K_{f}} , but each point on M_{K_{f}} will not necessarily come from such a piece on H^{g}(X_{0}(𝔠), ℚ).

5. The abelian variety associated to a Hilbert newform.

References Deligne [17] and Oda [40, Chap. II. sec. 1.].

Let A be a ring, E a free A-module and Q a quadratic form on E. We consider the subfunctor

F : A − Alg → Set

of B ↦ Hom_{A−mod}(E, B), where F (B) consists of all the A-linear maps u : E → B such that u(x)^{2} = Q(x), for all x ∈ E. We recall that the functor F is represented by the Clifford algebra we denote by C(E, Q) or C(E). We will denote by C^{+}(E, Q) or C^{+}(E) the even part of C(E, Q). For any bilinear symmetric form B on V, we will denote by C(E, B) or C(E) the Clifford algebra of the corresponding quadratic form.

Now, assume that A = k is a field of characteristic 0 and that Q is non-degenerate. Then E injects into C(E). Let CSpin(E) be the Clifford group. It is the algebraic group of all invertible elements g in C^{+}(E) such that gEg^{−1} = E. This gives a morphism φ of CSpin(E) onto SO(E) by letting φ(g) acts on E by φ(g) · x = gxg^{−1}. The kernel of φ is 𝔾_{m}, so one has an exact squence

0 ⟶ 𝔾_{m} ⟶^{w} CSpin(E) ⟶^{φ} SO(E) ⟶ 0.

Let C^{+}(E)^{×} be the algebraic group of all invertible elements in C^{+}(E). By definition, we have an inclusion

α : CSpin(E) ↪ C^{+}(E)^{×}.

This induces a representation of the algebraic group CSpin(E) on C^{+}(E) by:

g ∗ x = α(g) · x.

We denote this representation by C^{+}(E)_{s}. We let (C^{+}(E))_{ad} be the representation of CSpin(E) on C^{+}(E) given by

Ad(g) ∗ x = α(g) · x · α(g)^{−1}.

The action of CSpin(E) on C^{+}(E)_{s} is compatible with the right C^{+}(E)- module structure of C^{+}(E); and since End_{C^{+}(E)}(C^{+}(E)) consists of multiplication to the left by elements of C^{+}(E), one easily sees that the representation obtained on End_{C^{+}(E)}(C^{+}(E)) from C^{+}(E)_{s} is nothing else but C^{+}(E)_{ad}; that is, we have an isomorphism of representations:

End_{C^{+}(E)}(C^{+}(E)_{s}) = (C^{+}(E))_{ad}.

Also, from the structure theorem of Clifford algebras, we have an isomorphism of representations

(C^{+}(E))_{ad} = ⊕ ∧^{2i} E.

Now, let H_{ℤ} be a free ℤ-module of rank n + 2 endowed with a bilinear form of non-zero discriminant

B : H_{ℤ} ⊗ H_{ℤ} → ℤ.

5. THE ABELIAN VARIETY ASSOCIATED TO A HILBERT NEWFORM. 113

We assume that B has signature (n, 2). Let H_{ℚ} be a rational Hodge structure of weight 0 polarized by B and having Hodge numbers:

h^{−1, 1} = h^{1, −1} = 1, and h^{0, 0} = n.

Then, H_{ℚ} is given by a morphism of ℝ-groups h : \underline{S} → SO(H_{ℝ}), which lifts uniquely to a morphism h̃ : \underline{S} → CSpin(H_{ℝ}) fitting into the commutative diagram

𝔾_{m/ℝ} ⟶^{w}  \underline{S}  ⟶^{t} 𝔾_{m/ℝ}
∥  ↓h̃  ∥
𝔾_{m/ℝ} ⟶^{w} CSpin(H_{ℝ}) ⟶^{t} 𝔾_{m/ℝ}.

By Proposition 2.11 in Deligne [17], every representation of CSpin(H_{ℚ}) is polarizable with respect to h̃. As a consequence, one gets the following. Proposition 2.23. ([17, Deligne]) The Hodge structure of C^{+}(H_{ℚ})_{ad} defined by h̃ has type {(−1, 1), (0, 0), (1, −1)}. Proposition 2.24. [17, Proposition 4. 5] The Hodge structure of C^{+}(H_{ℚ})_{s} defined by h̃ has type {(0, 1), (1, 0)}. The Hodge structure C^{+}(H_{ℚ})_{s} together with the lattice C^{+}(H_{ℤ})_{s} is a polarized Hodge structure; therefore it defines an abelian variety. Remark 2.25. If we start with a rational Hodge structure H_{ℚ}, the resulting abelian variety is given up to ℚ-isogeny as this involves choosing a lattice H_{ℤ} in H_{ℚ} on which the restriction of B is integral valued.

Now, let us assume instead that we are given a Hodge structure H_{K} of weight 0 and rank n + 2 over the totally real number field K with a non–degenerate bilinear form

ψ : H_{K} ⊗ H_{K} → K.

Moreover, we assume that K acts on H_{K} as endomorphisms of Hodge structures. Then, we have

H_{K} ⊗_{ℚ} ℝ = ⨁_{i=1}^{g} H_{K} ⊗_{K, τ_{i}} ℝ,

and

H_{K} ⊗_{ℚ} ℂ = ⨁_{i=1}^{g} H_{K} ⊗_{K, τ_{i}} ℂ.

We assume that, for each i = 1, …, g, the extension of scalars ψ ⊗_{τ_{i}} ℝ of ψ has signature (n, 2) and defines a polarization of H_{K} ⊗_{K, τ_{i}} ℝ, and that the Hodge number of H_{K} ⊗_{K, τ_{i}} ℝ are

h^{1, −1} = h^{−1, 1} = 1 and h^{0, 0} = n.

By Definition 2.2, this datum defines a morphism of real algebraic groups

h : \underline{S} → SO(H_{K} ⊗_{ℚ} ℝ) = Res_{K/ℚ}(SO(H_{K}))_{ℝ}

of weight 0. By making use of the functoriality of Res_{K/ℚ}, one easily sees that h lifts to a unique morphism

h̃ : \underline{S} → CSpin(H_{K} ⊗_{ℚ} ℝ) = Res_{K/ℚ}(CSpin(H_{K}))_{ℝ} = G_{ℝ}

which fits in the commutative diagram

𝔾_{m/ℝ} ⟶^{w} \underline{S} ⟶^{t} 𝔾_{m/ℝ}
∥ ↓h̃ ∥
𝔾_{m/ℝ} ⟶^{w} G_{ℝ} ⟶^{t} 𝔾_{m/ℝ},

where G is the ℚ-algebraic group Res_{K/ℚ}(CSpin(H_{K})) and t ∘ w(x) = x^{−2}. The representations C^{+}(H_{K})_{s} and C^{+}(H_{K})_{ad} of CSpin(H_{K}) restrict functorially to ℚ-rational representations of the ℚ-group G on the ℚ-spaces Res_{K/ℚ}(C^{+}(H_{K})_{s}) and Res_{K/ℚ}(C^{+}(H_{K})_{ad}) respectively. Now, Proposition 4. 5 in [17] can be restated as follows.

5. THE ABELIAN VARIETY ASSOCIATED TO A HILBERT NEWFORM. 115

Proposition 2.26. [17, Proposition 4. 5] The Hodge structure of Res_{K/ℚ}(C^{+}(H_{K})_{s}) defined by h̃ is a polarized Hodge structure of type {(0, 1), (1, 0)}.

By choosing an O_{K}-lattice H_{O_{K}} of H_{K} to which the restiction of ψ is O_{K}-valued, we get a lattice Res_{O_{K}/ℤ}(C^{+}(H_{O_{K}})_{s}) in Res_{K/ℚ}(C^{+}(H_{K})_{s}). This then defines a polarized abelian variety. Any other choice of lattice gives an F -isogenous variety.

Applying this construction to the Hodge structure H^{2}(M_{f}, ℚ)(1), we get a Hilbert-Blumenthal abelian variety A_{f}. We call it the Hilbert-Blumenthal abelian variety attached to f. By making use of the fact that Ψ_{f} is a kernel form, one can be a bit more specific about A_{f}.

Theorem 2.27. [40, sec. 5.5] The K_{f}-rational Hodge structure H^{2}(M_{f}, ℚ) is the tensor product of two K_{f}-rational Hodge structures of weight 1:

H^{2}(M_{f}, ℚ) = W^{(1)} ⊗_{K_{f}} W^{(2)}.

Therefore the variety A_{f} is of the form A_{f} = A_{f}^{1} × A_{f}^{1} × A_{f}^{2} × A_{f}^{2}.

The period moduli of those varieties are given by the following result of Oda.

Theorem 2.28. [40, Theorem 6.6] Let f ∈ S_{2}(𝔠) be a normalized eigenform, and f^{σ} the companions of f, where σ runs over all the real embeddings of K_{f}. Choose a canonical basis of H^{2}(M_{f}, ℚ) and form the periods

Ω_{ss′}(f^{σ}).

Then, the period moduli of the two Hilbert-Blumenthal abelian varieties A_{f}^{1} and A_{f}^{2} are given by the points τ_{1}, τ_{2} ∈ (ℂ − ℝ)^{d} given by

τ_{1} = (Ω_{−+}(f^{σ})/Ω_{++}(f^{σ}))_{σ}, τ_{2} = (Ω_{+−}(f^{σ})/Ω_{++}(f^{σ}))_{σ}.

6. Computations.

In this section, we explain, in a specific example, how one can compute the period modulus of the Hilbert-Blumenthal abelian variety associated to a newform f ∈ S_{2}(𝔠). One way of doing this is to exploit the relation given by Oda [40] between such periods and special values of twists of L–functions. Those methods can then be applied easily when p_{g}(X_{0}(𝔠)) = 1.

In the rest of this section, we let F = ℚ(\sqrt{5}) and ω = (1 + \sqrt{5})/2. Let 𝔫 = (ν) be an integral ideal and V ⊂ O_{F}^{×}^{+} a subgroup of finite index such that V ⊂ 1 + 𝔫. For any α ∈ O_{F}, we define

D_{α/ν} = {z ∈ H^{2} : Re(z) = α/ν}.

We have a map i : V \D_{α/ν} → Γ\H^{2}, where V acts on D_{α/ν} by z ↦ εz +(1−ε)α/ν. The quotient V \D_{α/ν} is topologically a cylinder S^{1}×ℝ and its image in X_{Γ} can be compactified by adding the orbits of α/ν and ∞ in P^{1}(F). This gives a 2-cycle γ_{α/ν, V} in X_{Γ}. (Replacing V by a smaller subgroup V′ multiplies this cycle by [V : V′]). The following lemma is from Oda [40].

Lemma 2.29. [40, Lemma 15.2] If χ : (O_{F} /𝔫)^{×} → ℂ^{×} is a non-trivial character, then

γ_{χ, V} = ∑_{α∈O_{F} /𝔫}χ(α)γ_{α/ν, V} ∈ Im(H_{2}(X′_{Γ}, ℚ) → H_{2}(X_{Γ}, ℚ)).

By projection, γ_{χ, V} defines a class in H_{2}^{sp}(X_{Γ}, ℚ)).

Lemma 2.30. [40, Lemma 15.4] Let χ : (O_{F} /𝔫)^{×} → ℂ^{×} be a primitive character. Then, we have

γ_{χ, V} ∈ \{ \begin{matrix}H_{2}^{sp}(X_{Γ}, ℚ))_{++} ⊕ H_{2}^{sp}(X_{Γ}, ℚ))_{−−} & if χ(−1) = 1, \\ H_{2}^{sp}(X_{Γ}, ℚ))_{+−} ⊕ H_{2}^{sp}(X_{Γ}, ℚ))_{−+} & if χ(−1) = −1. \end{matrix}

Moreover, γ_{χ, V} ∈ H_{2}^{sp}(X_{Γ}, ℚ))_{ss′} if χ(ω) = s and χ(ω̄) = s′.

Oda had shown that integrating along the cycle γ_{χ, V}, one gets special values of twisted L-functions.

Theorem 2.31. [40, Theorem 16.3] Let f ∈ S_{2}(𝔠) be a normalized eigenform. Then

L(f, 1, χ) = \frac{1}{[O_{F}^{×}^{+} : V]} \frac{1}{(2π)^{2}} \frac{1}{G(χ̄)} ∫_{γ_{χ̄, V}} ω_{f},

where V is of sufficiently large index and the orientation of γ_{χ̄, V} is appropriately chosen, and G(χ) the Gauss sum

G(χ) = ∑_{α∈O_{F} /(ν)} χ(α) exp(2πTr(ωα/ν\sqrt{D})).

This theorem gives us a way to compute the period moduli of the abelian varieties A_{f}^{1} and A_{f}^{2}. Let us fix a normalized eigenform f ∈ S_{2}(𝔠) and a primitive character χ of conductor 𝔫 = (ν). We let W_{A} be the Atkin-Lehner involution given by

W_{A} : z = (z_{1}, z_{2}) ↦ (−\frac{1}{Az_{1}} , −\frac{1}{Āz_{2}} ),

where A is a totally positive generator of 𝔠. Let

f(z_{1}, z_{2}) = ∑_{μ∈O_{F}^{+}} c((μ)) exp[2πi(Tr(\frac{μωz}{\sqrt{D}})]
= ∑_{μ∈O_{F}^{+}/O_{F}^{×}^{+}} c((μ)) ∑_{ε∈ O_{F}^{×}^{+}} exp[2πiTr(\frac{εμωz}{\sqrt{D}})]

be the q-expansion of f. Then f_{χ} = f ⊗ χ ∈ S_{2}(𝔠𝔫^{2}) and its q-expansion is given by

f ⊗ χ(z_{1}, z_{2}) = ∑_{μ∈O_{F}^{+}} c((μ))χ(μ) exp[2πiTr(\frac{μωz}{\sqrt{D}})]
= ∑_{μ∈O_{F}^{+}/O_{F}^{×}^{+}} c((μ))χ(μ) ∑_{ε∈ O_{F}^{×}^{+}} exp[2πiTr(\frac{εμωz}{\sqrt{D}})].

Lemma 2.32. Let f ∈ S_{2}(𝔠) be an eigenform. If W_{A}f = −f, then L(f) = 0; otherwise

L(f, 1) = −\frac{D}{2π^{2}} ∑_{μ∈O_{F}^{+}} \frac{c((μ))}{N(μ)} [1 − exp (−\frac{2πμω}{\sqrt{DA}})] ×

exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄ \\ √ & \overline{Ā} \end{matrix} − \frac{μ}{\sqrt{A}})] .

Proof. By definition,

L(f, 1) = ∫ ∫_{O_{F}^{×}^{+}\ℝ_{+}^{2}} f(iy_{1}, iy_{2})dy_{1}dy_{2} = ∫_{τ_{0}}^{ω^{2}τ_{0}} ∫_{0}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2},

with τ_{0} arbitrarly chosen on the imaginary axis. But choosing τ_{0} = i/ω\sqrt{A}, and making the change of variable z ↦ W_{A}(z), we get

L(f, 1) = ∫_{τ_{0}}^{ω^{2}τ_{0}} ∫_{0}^{i∞} f||W_{A}(z_{1}, z_{2})dz_{1}dz_{2}
= ε_{A}L(f, 1).

This gives the first part of the lemma. To get the second one, keeping the same choice of τ_{0} as before, we choose C on the imaginary axis and split the integral as

L(f, 1) = ∫_{τ_{0}}^{ω^{2}τ_{0}} ∫_{0}^{C} f(z_{1}, z_{2})dz_{1}dz_{2} + ∫_{τ_{0}}^{ω^{2}τ_{0}} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2}.

Again choosing C = i/\sqrt{Ā} and making the change of variable z ↦ W_{A}(z) in the first integral of the sum, we get that

L(f, 1) = (1 + ε_{A}) ∫_{τ_{0}}^{ω^{2}τ_{0}} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2}.

We then conclude the proof of the lemma by an integration term by term. □

Remark 2.33. Let f ∈ S_{2}(𝔠) be an eigenform with W_{A}f = ε_{A}f, and χ a quadratic character of conductor 𝔫 = (ν) with (𝔠, 𝔫) = 1. When χ is a quadratic character, f_{χ} ∈ S_{2}(𝔠𝔫^{2}) and W_{Aν^{2}}f_{χ} = ε_{A}χ(−A)f_{χ}. Therefore, by the above Lemma, when ε_{A}χ(−A) = 1,

L(f, 1, χ) = −\frac{D}{2π^{2}} ∑_{μ∈O_{F}^{+}} \frac{c((μ))}{N(μ)}χ(μ) [1 − exp (−\frac{2πμω}{ν\sqrt{DA}})] ×

exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄ \\ ν̄ & \sqrt{Ā} \end{matrix} − \frac{μ}{ν\sqrt{A}})] .

By using the fact that every totally positive element is of the form ω^{2k}, k ∈ ℤ, this series can be rearranged as

L(f, 1, χ) =

− \frac{D}{2π^{2}} ∑_{μ∈O_{F}^{+}/O_{F}^{+}^{×}} \frac{c((μ))}{N(μ)}χ(μ) × ∑_{k∈ℤ}[1 − exp (−\frac{2πμω^{2k+1}}{ν\sqrt{DA}} )] ×

exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄^{2k+1} \\ ν̄ & \sqrt{Ā} \end{matrix} − \frac{μω^{2k}}{ν\sqrt{A}})] .

When χ is not a quadratic character, we need to resort to a different approach to express L(f, 1, χ) as a series. This involves the calculation of the partial L-functions that appear in Oda’s Theorem. As we will see shortly, this has the inconvenient of bringing some extra factor in the denominator of the exponential sum with the bad effect of slowing down the convergence of the series. To start, we recall that

f_{χ} = G(χ̄)^{−1} ∑_{α∈O_{F} /𝔫} \overline{χ(α)}f||γ_{α, ν} with γ_{α, ν} = (\begin{matrix}ν & α \\ 0 & ν \end{matrix}) ,

so that

L(f, 1, χ) = G(χ̄)^{−1} ∑_{α∈O_{F} /𝔫} \overline{χ(α)}L(f||γ_{α, ν}, 1).

The partial L-functions are then computed using the following procedure.

Lemma 2.34. Let f be a normalized eigenform, with f||W_{A} = ε_{A}f. We let P = {L(f||γ_{α, ν}, 1), α ∈ (O_{F} /𝔫)^{×}} be the set of all partial Lfunctions that contribute to L(f, 1, χ), and w_{A} : (O_{F} /𝔫)^{×} → (O_{F} /𝔫)^{×} the isomorphism α ↦ −1/Aα. Then the map L(f||γ_{α, ν}, 1) ↦ L(f||γ_{w_{A}(α), ν}, 1) is a permutation of P. Moreover, L(f||γ_{α, ν}, 1) = ε_{A}L(f||γ_{w_{A}(α), ν}, 1) for all α, and

L(f||γ_{α, ν}, 1) = −\frac{D}{(2π)^{2}} ∑_{μ∈O_{F}^{+}} \frac{c((μ))}{N(μ)} [1 − exp (−\frac{2πμω}{ε_{0}ν\sqrt{DA}}(ε_{0}^{2} − 1))] ×

[exp (−2πiTr (\frac{μωα}{ν\sqrt{D}})) + ε_{A} exp (−2πiTr (\frac{μωw_{A}(α)}{ν\sqrt{D}}))] ×

exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄ \\ ν̄ & \sqrt{Ā} \end{matrix} − \frac{μω}{ε_{0}ν\sqrt{A}})] ,

where ε_{0} ∈ O_{F}^{×} is chosen such that ε_{0} > 0 and ε_{0}^{2} − 1 ∈ (ν).

Proof. Let β ∈ O_{F} be such that Aαβ = −1 mod (ν). Then,

there exists γ_{1} = (\begin{matrix}ν & α \\ Aβ & d \end{matrix}) ∈ Γ_{0}(𝔠) such that γ_{1}(−α/ν) = 0. Take γ_{0} = γ_{α, ν}^{−1} (\begin{matrix}ε_{0}^{2} & 0 \\ 0 & 1 \end{matrix}) γ_{α, ν} in Γ_{0}(𝔠) and put γ = γ_{1}γ_{0}. Then, by definition,

L(f||γ_{α, ν}, 1) = ∫ ∫_{V \D_{α, ν}} f(z_{1}, z_{2})dz_{1}dz_{2}
= ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{−ᾱ/ν̄}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2},

where we choose τ_{0} ∈ H with Re(τ_{0}) = −α/ν. By making the substitution z ↦ γ_{1}^{−1}W_{A}(z), we get that

L(f||γ_{α, ν}, 1) = ∫_{−1/Aγ(τ_{0})}^{−1/Aγ_{1}(τ_{0})} ∫_{−β̄/ν̄}^{i∞} f||W_{A}(z_{1}, z_{2})dz_{1}dz_{2}.

Taking τ_{0} = −α/ν + i/ε_{0}ν\sqrt{A}, this gives

L(f||γ_{α, ν}, 1) = ∫_{τ_{1}}^{γ′_{0}(τ_{1})} ∫_{−β̄/ν̄}^{i∞} f||W_{A}(z_{1}, z_{2})dz_{1}dz_{2},

where τ_{1} = −β/ν + i/ε_{0}ν\sqrt{A} and γ′_{0} = γ_{β, ν}^{−1} (\begin{matrix}ε_{0}^{2} & 0 \\ 0 & 1 \end{matrix}) γ_{β, ν}. This just

says that

L(f||γ_{α, ν}, 1) = ε_{A}L(f||γ_{β, ν}, 1)
= ε_{A}L(f||γ_{w_{A}(α), ν}, 1),

and the first part of the lemma then follows. To the second one, we take C ∈ H with Re(C) = −ᾱ/ν̄, and split the integral as

L(f||γ_{α, ν}, 1) = ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{−ᾱ/ν̄}^{C} f(z_{1}, z_{2})dz_{1}dz_{2}
+ ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2}
= ∫_{γ_{1}(τ_{0})}^{γ_{1}γ_{0}(τ_{0})} ∫_{0}^{γ̄_{1}(C)} f(z_{1}, z_{2})dz_{1}dz_{2}
+ ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2},

122 2. SOME NUMERICAL RESULTS ON THE ODA CONJECTURE.

where the second equality is obtained by making the change of variable z ↦ γ_{1}^{−1}(z) in the first integral and using the Γ_{0}(𝔠)-invariancy of f. The substitution z ↦ W_{A}(z) then gives:

L(f||γ_{α, ν}, 1) = ∫_{−1/Aγ(τ_{0})}^{−1/Aγ_{1}(τ_{0})} ∫_{−1/Āγ̄_{1}(C)}^{i∞} f||W_{A}(z_{1}, z_{2})dz_{1}dz_{2}
+ ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2}.

Again, choosing τ_{0} = −α/ν + i/ε_{0}ν\sqrt{A} and C = −ᾱ/ν̄ + i/ν̄\sqrt{Ā}, we obtain that

L(f||γ_{α, ν}, 1) = ε_{A} ∫_{τ_{1}}^{γ′_{0}(τ_{1})} ∫_{−1/Āγ̄_{1}(C)}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2}
+ ∫_{τ_{0}}^{γ_{0}(τ_{0})} ∫_{C}^{i∞} f(z_{1}, z_{2})dz_{1}dz_{2},

with Im(τ_{0}) = Im(τ_{1}) = 1/ε_{0}ν\sqrt{A}, Im(γ_{0}(τ_{0})) = Im(γ′_{0}(τ_{1})) = ε_{0}/ν\sqrt{A} and Im(C) = Im(−1/Āγ̄_{1}(C)) = 1/ν̄\sqrt{Ā}. We then complete the lemma by integrating term by term. □

By making use of the identity

\overline{χ}(μ)G(χ) = ∑_{α∈O_{F} /𝔫}χ(α) exp (2πiTr (\frac{μωα}{\sqrt{D}}))

and the fact that \overline{χ}(−Aαβ) = 1, we see that

∑_{α∈O_{F} /(ν)} \overline{χ}(α) [exp (−2πiTr (\frac{μωα}{ν\sqrt{D}}))

+ ε_{A} exp (−2πiTr (\frac{μωw_{A}(α)}{ν\sqrt{D}}))]

= χ(μ)G(\overline{χ}) + ε_{A}χ(−A)\overline{χ}(μ)G(χ).

Thus reversing the order of the summation in L(f, 1, χ), we get the following proposition.

Proposition 2.35. Let f ∈ S_{2}(𝔠) be an eigenform with W_{A}f = ε_{A}f, and χ a character of conductor 𝔫 = (ν) with (𝔠, 𝔫) = 1. Then,

L(f, 1, χ) = S(f, 1, χ) + ε_{A}χ(−A)\frac{G(χ)}{G(\overline{χ})}S(f, 1, \overline{χ}),

where

S(f, 1, χ) = −\frac{D}{(2π)^{2}} ∑_{μ∈O_{F}^{+}} \frac{c((μ))}{N(μ)}χ(μ)×

[1 − exp (−\frac{2πμω}{ε_{0}ν\sqrt{DA}}(ε_{0}^{2} − 1))] exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄ \\ ν̄ & \sqrt{Ā} \end{matrix} − \frac{μω}{ε_{0}ν\sqrt{A}})] .

Remark 2.36. Again making use of the fact that every totally positive element is of the form ω^{2k}, k ∈ ℤ, we get

S(f, 1, χ) = −\frac{D}{2π^{2}} ∑_{μ∈O_{F}^{+}/O_{F}^{×}^{+}} \frac{c((μ))}{N(μ)}χ(μ)×

∑_{k∈ℤ}[1 − exp (−\frac{2πμω^{2k+1}}{ε_{0}ν\sqrt{DA}} (ε_{0}^{2} − 1))] ×

exp [\frac{2π}{\sqrt{D}} (\begin{matrix}μ̄ & ω̄^{2k+1} \\ ν̄ & \sqrt{Ā} \end{matrix} − \frac{μω^{2k+1}}{ε_{0}ν\sqrt{A}})] .

Remark 2.37. The rate of convergence of those two series is controlled by the exponential sum. So to improve the convergence of our algorithm we must choose the character χ with conductor as small as possible. This turns out to be very delicate as a satisfactory error term requires to compute all Hecke operators for primes 𝔭 such that the ratio N(𝔭)/DN(𝔠𝔫^{2}) is large enough.

Example 2.38. For 𝔠 = 31A = (5+2ω), we take 𝔫_{1} = (3) and 𝔫_{2} = (4). Then (O_{F} /𝔫_{1})^{×} = 𝔽_{9}^{×} = ⟨ω⟩ and (O_{F} /𝔫_{2})^{×} = ⟨−1, ω⟩. We let χ_{1} be the unique quadratic character of conductor 𝔫_{1} and χ_{2} the quadratic character given by

χ_{2}(ω) = −1 = χ_{2}(−1).

Then γ_{χ_{1}, V_{1}} ∈ H_{2}(X_{0}(𝔠), ℚ)_{−−} with V_{1} = ⟨ω^{8}⟩ and γ_{χ_{2}, V_{2}} ∈ H_{2}(X_{0}(𝔠), ℚ)_{−+} with V_{2} = ⟨ω^{6}⟩. We complete γ_{χ_{1}, V_{1}} and γ_{χ_{2}, V_{2}} to get a canonical basis. Then, Ω_{++} and Ω_{+−} are completely determined by the Riemann–Hodge relation. We get

Ω_{−−} = −67.464569223618835646128485047426897

and

Ω_{−+} = −128.75794400309976871557728718976371i.

This gives

τ′ = −Ω_{−−}/Ω_{−+} = 0.52396432504385645846658571918312i

The curve E/ℚ(\sqrt{5}) : y^{2} + xy + ωy = x^{3} − (1 + ω)x^{2} has conductor 31A = (5 + 2ω). The PARI procedure on elliptic curves yields the following periods for this curve:

ω_{1} = 3.0521731533572582695976003439575002

and

ω_{2} = 2.39884476932372222732017398860862i.

We get

τ = \frac{ω_{2}}{ω_{1}} = 0.78594648756578468769989532640147425i.

The ratio

\frac{τ}{τ′} = 1.50000000000000000000003196

is close to 3/2 with 21 decimals accuracy. (Note that this is consistent with the fact that E has some 2–torsion).

Example 2.39. For 𝔠 = 41A = (6 + ω), we take 𝔫 = (4). Then, (O_{F} /𝔫)^{×} = ⟨−1, ω⟩. We let χ_{1} the quadratic character given by

χ_{1}(ω) = −1 = χ_{1}(−1),

and χ_{2} the quadratic character given by

χ_{2}(ω) = −1 and χ_{2}(−1) = 1.

6. COMPUTATIONS. 125

Then γ_{χ_{1}, V} ∈ H_{2}(X_{0}(𝔠), ℚ)_{−+} and γ_{χ_{2}, V} ∈ H_{2}(X_{0}(𝔠), ℚ)_{−−} with V = ⟨ω^{6}⟩. We complete γ_{χ_{1}, V} and γ_{χ_{2}, V} to get a canonical basis. We get

Ω_{−−} = −54.40018580695956431139383963207

and

Ω_{−+} = −160.12204086937231354613169017024i.

This gives

τ′ = −Ω_{−−}/Ω_{−+} = 0.3397420212207966955319258264730i.

The curve E/ℚ(\sqrt{5}) : y^{2} + ωy = x^{3} − ωx^{2} has conductor 41A = (6 + ω). Again, the PARI procedure yields the real and imaginary periods

ω_{1} = 7.227157478384688293199095999068

and

ω_{2} = 2.455369089387410305504378228262i.

This gives

τ = \frac{ω_{2}}{ω_{1}} = 0.33974202122079669553215993921i.

The ratio

\frac{τ}{τ′} = 1.000000000000000000000689

is close to 1 with 22 decimals accuracy.

7. Program Listing

We use this program to compute the Oda periods.

\\THIS ROUTINE COMPUTE THE ODA PERIODS.
\\Initializing parameters.
w=quadgen(5);
field=nfinit(z^2-z-1);
fieldembed(x)=Mod(real(x)+imag(x)*z, z^2-z-1);
level=6+w;
read(uniformizers1);
read(41Acoeffs2_10979);
\\This function returns the primes in $O_K$ lying above\\
\\any prime $p$ in $\Z$.
gen(pr, res)=res=0;\
   if(kronecker(pr,5)==-1,res=pr,N0=floor(sqrt(pr-1));\
                                   a=N0;res1=r^2-a*r+pr-a^2;\
       while(polisirreducible(res1),a=a-1;res1=r^2-a*r+pr-a^2);\
            res=floor(real(polroots(res1)))~;\
     if(res[1]==res[2], res=a+res[1]*w, res=[a,a]+w*res)); res;

\\Creating the group of unit in $O_K/I$ for some ideal $I$. We are only
\\interested in the reduction of the torsion free part. We call $nu$ a
\\generator of $I$.
\\nu=w^10-1;
nu=3;
nu1=[]; res=idealfactor(field, fieldembed(nu));\
for(j=1, matsize(res)[1], res0=nfmodprinit(field, res[j,][1]);\

            nu1=concat(nu1, [res0])); nu1;
nu0=vector(length(nu1), i1, 0); for(j=1, length(nu1),\
                                res=idealnorm(field, nu1[j]);\
       if(issquare(res)==1, nu0[j]=sqrtint(res), nu0[j]=res)); nu0;
nu2=3;
nu3=4;

G=vector(3);\
G[1]=1; for(j=1, length(nu0), if(kronecker(nu0[j],5)==-1,\
             G[1]=G[1]*(nu0[j]^2-1), G[1]=G[1]*(nu0[j]-1))); G[1];\
G[2]=[];\
for(j=1, length(nu0), if(kronecker(nu0[j],5)==-1, res=nu0[j]^2-1,\
                     res=nu0[j]-1); G[2]=concat(G[2], res)); G[2];\
G[3]=[];\
for(j=1, length(G[2]), s=divisors(G[2][j]);\
if(kronecker(nu0[j],5)==-1, i=0; k=1; n=1;\
                        res=Mod(i, nu0[j])+Mod(k, nu0[j])*w;\
  until(n==length(s), n=1; k=k+1; if(k>=nu0[j], k=1; i=i+1);\
                          res=Mod(i,nu0[j])+Mod(k, nu0[j])*w;\
     while((!(res^s[n]==Mod(1, nu0[j])))&&n<length(s), n=n+1));\
       res=lift(res); G[3]=concat(G[3], res),\
     l=1; until(n==length(s), n=1; l=l+1; res=Mod(l, nu0[j]);\
     while((!(res^s[n]==Mod(1, nu0[j])))&&n<length(s), n=n+1));\
         res=lift(res); G[3]=concat(G[3], res))); print(G);
G0=vector(3);\
G0[1]=G[1]; G0[2]=G[2]; G0[3]=[]; for(j=1, length(G[2]),\
  if(!(imag(G[3][j])==0), res=Mod(real(G[3][j]), nu0[j])+Mod(imag(G[3][j]),\
               nu0[j])*w, res=Mod(G[3][j], nu0[j]));\
                G0[3]=concat(G0[3], res)); print(G0);

\\In the cyclic group of order $G[2][j]$, this function returns the
\\order of any element $g$.
ord(g, j, s)=s=1; divs=divisors(G[2][j]); k=1;\

g0=nfeltreducemodpr(field, fieldembed(g), nu1[j]);\
     g0=g0[1]+g0[2]*w;\
     while((!(g0^divs[k]==Mod(1, nu0[j])))&&k<length(divs), k=k+1);\
          s=divs[k]; s;
\\Writing any element of the local factor at the j^th primes as \\
\\a power of the cyclic generator $G[3][j]$.
decomp(g, j, res0)=res0=1; res1=G[2][j]/ord(g,j); i=0;\
g0=nfeltreducemodpr(field, fieldembed(g), nu1[j]); g0=g0[1]+g0[2]*w;\
  while(!(G0[3][j]^i==g0), i=i+1); res0=i; res0;

\\ This function returns the quadratic residue character modulo $\nu$.
\\We first define the local residue character at each prime. The global
\\character is just the product of those one.
quad0(g, j, res)=res=1; res1=(G[2][j]/ord(g,j))%2; res=(-1)^res1; res;
quad(g, res0)=res0=0;\
  if(!(idealadd(field, fieldembed(g), fieldembed(nu))==[1, 0; 0, 1]),\
        res0=0, res0=1; for(j=1, length(G[2]), res0=res0*quad0(g, j))); res0;
print([quad(-1), quad(w), quad(1-w), quad(-level)]);

\\Here we create the quadratic character over $O_K/(4)$. We use the fact that
\\$O_K/(4)^*$ is generated by $w$ and $-1$.
quad2(g, res3)=res3=0;\
  if(!(idealadd(field, fieldembed(g), [4, 0; 0, 4])==[1, 0; 0, 1]),\
    res3=0, res3=1; if(g==-1||g==w, res3=-1, res0=g%4;\
    j=1; until((w^j%4==res0||-w^j%4==res0), j=j+1);\
      if(w^j%4==res0, res3=(-1)^j, res3=(-1)^(j+1)))); res3;
print([quad2(-1), quad2(w), quad2(1-w), quad2(-level)]);

quad3(g, res3)=res3=0;\

  if(!(idealadd(field, fieldembed(g), [4, 0; 0, 4])==[1, 0; 0, 1]),\
    res3=0, res3=1; if(g==-1||g==w, if(g==w, res3=-1, res3=1), res0=g%4;\
    j=1; until((w^j%4==res0||-w^j%4==res0), j=j+1); res3=(-1)^j)); res3;
print([quad3(-1), quad3(w), quad3(1-w), quad3(-level)]);

\\This function gives the factorization of all ideals of norm less than
\\$N$ in terms of uniformizer given by the gen function above.
N=10979;
ideals=[];\
ids=ideallist(field, N);\
ids0=[]; for(j=2, length(ids), ids0=concat(ids0, ids[j]));\
for(j=1, length(ids0), res0=idealfactor(field, ids0[j]);\
              res3=matrix(matsize(res0)[1], 2);\
for(k=1, matsize(res0)[1], res2=res0[k,1][1];\
 if(kronecker(res2,5)==-1, res3[k,]=[res2, res0[k,2]],\
  if(kronecker(res2,5)==0, res3[k,]=[gen(5), res0[k,2]],\
    if(kronecker(res2,5)==1 &&\
             idealfactor(field, fieldembed(gen(res2)[1]))[1,1]==res0[k,1],\
         res3[k,]=[gen(res2)[1], res0[k,2]], res3[k,]=\
                  [conj(gen(res2)[1]), res0[k,2]]))));\
            ideals=concat(ideals, [res3])); ideals;

\\For an ideal given in the above form, this function returns a generator.
idgen(id, res4)=res4=1;\
 for(k1=1, matsize(id)[1], res4=res4*id[k1,][1]^id[k1,][2]); res4;

\\This function returns the Hecke eigenvalue associated to any prime\\
power given in the above list.
hcoeff0(id0,res0)=res0=1;\
 j1=1; while(!(id0[1]==unifs[j1]), j1=j1+1); j1;\

   if(denominator(level/unifs[j1])==1, res0=ap[j1]^id0[2],\
     if(id0[2]==1, res0=ap[j1], res0=res2=ap[j1]; res1=1;\
       for(k2=2, id0[2], res0=ap[j1]*res2-norm(unifs[j1])*res1;\
                                  res1=res2; res2=res0))); res0;

\\This function compute the Hecke eigenvalue associated to any ideal\\
\\given in the the list above.
hcoeff(id, res5)=res5=1;\
  for(l=1, matsize(id)[1], res5=res5*hcoeff0(id[l,])); res5;

\\This compute the exponential sum corresponding to the residue
\\$\alpha$ modulo $\nu$.
subsum(id1, nus, sum0)=sum0=0;\
M=[0,0]; M0=-1;\
  while(abs(2*Pi*conj(idgen(id1))*(1-w)^(2*M0+1)/(conj(nus)*(2*w-1)*\
      sqrt(conj(level))))<=10^5, M0=M0-1); M[1]=M0;\
M0=1; while(2*Pi*idgen(id1)*w^(2*M0+1)/(nus*(2*w-1)*\
                     sqrt(level))<=10^5, M0=M0+1); M[2]=M0;\
sum0=sum(k3=M[1], M[2], (1-exp(-2*Pi*idgen(id1)*\
                w^(2*k3+1)/(nus*(2*w-1)*sqrt(level))))*\
   exp((2*Pi/(2*w-1))*(conj(idgen(id1)*w^(2*k3+1))/(conj(nus)*\
        sqrt(conj(level)))- idgen(id1)*w^(2*k3)/(nus*sqrt(level))))); sum0;

\\This returns an approximation of the L-function...???

L0=res6=subsum(Mat([1,1]), nu3);\
    res6=res6+sum(m=1, length(ideals), hcoeff(ideals[m])*\
        quad3(idgen(ideals[m]))/norm(idgen(ideals[m]))*\
             subsum(ideals[m], nu3)); L0=-res6;
print(L0);

Wmm=sqrt(abs(norm(nu)))*L0; print(Wmm);

L=res7=subsum(Mat([1,1]), nu3);\
    res7=res7+sum(m=1, length(ideals),\
          (hcoeff(ideals[m])/norm(idgen(ideals[m])))*\
              quad2(idgen(ideals[m]))*subsum(ideals[m], nu3)); L=-res7;
print(L);
Wmp=sqrt(abs(norm(nu2)))*L; print(Wmp);

8. Conclusion.

The theory of algebraic modular forms in Gross [29] gives a very nice way to compute Hilbert modular eigenforms of weight 2 over F = ℚ(\sqrt{5}) by transferring such computations to the Hamilton quaternion algebra by means of the Jacquet–Langlands correspondence. The same approach can be used over any totally real field F (of narrow class number one) and for more general (including non-parrallel) weights (see the recent paper of Consani and Scholten [12]). The major requirement of the algorithms we have used is finding an efficient way to compute the Hecke operators. This becomes an imperative issue as the norm of the level increases, in order to get a satisfactory precision on the period moduli of the computed newforms. So it would be interesting to see wether or not the icosian ring admits a Euclidean algorithm. If affirmative, such a result will have the dramatic effect of increasing the efficiency of our algorithm. This is an issue that we intend to explore.

In contrast to the one dimensional case, the computation of periods in higher dimension seems to be a delicate matter. Indeed, from the examples we discussed, the computation of the Oda periods requires a large number of Hecke eigenvalues. Indeed, for the two smallest primes in our list, namely 31 and 41, we needed the Hecke eigenvalues for all primes of norm less than or equal to 10000. But this could only give us an accuracy of no more than 20 decimals; and this does not seem to be enough to determine either an integral or a rational model in the corresponding isogeny classes. The main difficulty lies in the fact that the computation of the Oda periods implicitly involves the choice of a lattice in the rational cohomology group H^{2}(M_{f}, ℚ) associated to the

8. CONCLUSION. 133

newform f, and this is a big difference with the one dimensional case. Indeed, when F = ℚ, not only is it known that the elliptic curve E_{f} is rational, but also a result of Edixhoven [21] on the Manin constant, says that the modular symbols computations, as done in Cremona [14], directly leads to an integral model. This greatly cuts down the amount of computations needed in this case. The modular parametrization of elliptic curves play a rôle in the definition of the Manin constant. In our setting, one is then tempted to consider its natural analogue which is the parametrization of elliptic curves by Shimura curves. Due to the lack of q–expansion in this situation, one then asks for the right substitute to the Manin constant and how this will ensure us with a good model for the isogeny class of the elliptic curve corresponding to our newform. There is no doubt that this question is related to the one of accessing the fundamental periods of the newform f. As this is a very active field in the theory of Hilbert modular forms, the few results in our second chapter should be seen more as a work in progress. We would like to explore the issue of overcoming the torsion in H^{2}(M_{f}, ℚ) in order to obtain an integral model for the computed isogeny class. We would also like to extend the PARI search that we started with the help of Prof. N. Elkies and see if we can make use of results in the spirit of Faltings–Serre or the more recent results of Skinner–Wiles to prove the modularity of the curves we have obtained (and will obtain).

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