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\begin{center}
{\Huge 189-726B: Modular Forms II} \\ \sk
{\Huge Assignment 2} \\ \sk
{\Large Due: Wednesday, January 23 }
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\noindent
1. Prove the recurrence formula
$$ {\tilde T}_{p^{r-1} }
{\tilde T}_{p}([\Lambda]) =
{\tilde T}_{p}
{\tilde T}_{p^{r-1} }([\Lambda]) =
{\tilde T}_{p^{r}}([\Lambda]) + p {\tilde T}_{p^{r-2}}([p\Lambda])
$$
for the abstract Hecke operators on lattices that
was used in class on Monday.
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\noindent
2. Show that the Hecke operators $T_n$ acting on the space $S_k(\Gamma_0(N))$
are self-adjoint with respect to the Petersson scalar product,
when $\gcd(n,N)=1$. You may eventually have to extend some definitions
that were only given for $\SL_2(\Z)$. Try to use only your notes, and avoid
peeking into a textbook!
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3. Show that the modular forms $f(\tau)$ and $f(d'\tau)$ belong to
$S_k(\Gamma_0(N))$ when $dd'$ divides $N$ and $f$ belongs
to $S_k(\Gamma_0((d))$. Show that if $f$ is an eigenvector for $T_n$ with
$\gcd(n,N)=1$, then the same holds for $f(d'\tau)$, and that the eigenvalues
are equal.
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4. Let ${\bf T}$ be the $\Q$-algebra of Hecke operators $T_n$ acting on
$S_{24}(\SL_2(\Z))$. Express ${\bf T}$ as a product of
specific number fields, and
write down a basis of eigenforms for $S_{24}(\SL_2(\Z))$.
(This fun calculation was done by Hecke.)
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