189-726A: L-functions and Modular Forms Assignment 3

Due: Wednesday, October 19

1. Let w_{N} be the Fricke involution that was introduced in class, acting on modular forms of weight k.

a) Show that, if f belongs to S_{k}(Γ_{0}(N), χ), then w_{N}f belongs to S_{k}(Γ_{0}(N), χ̄).

b) Conclude that, if f is a newform, then there is a complex scalar w_{f} of norm one such that

w_{N}f = w_{f} · f̄,

where f̄ denotes the modular form obtained from f by conjugating its fourier expansion.

c) Use the result of part b) to write down a functional equation relating L(f, s) to L(f̄, s).

2. Let T be the Q-algebra of Hecke operators T_{n} acting on S_{24}(SL_{2}(Z)). Express 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.)