3
votes
1answer
116 views

modular form -Petersson inner product

my question is about Petersson inner product. i need to prove that $(E_k,f) =0 $ $\forall f \in S_k(SL_2(\mathbb{Z}))$ the only thing that i think that should help me is that the space of cusp form ...
7
votes
2answers
447 views

Ramanujan's Tau function, an arithmetic property

The problem: Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ ...
9
votes
1answer
310 views

Relation between congruence subgroups. $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$

I'm hoping there's a pleasant way to solve this one. Prove that $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$. Showing that $\Gamma(M)\Gamma(N) \subset \Gamma(\gcd(M,N))$ is rather straight forward, ...
8
votes
1answer
180 views

Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} ...
3
votes
1answer
158 views

Bounds for Fourier coefficients of cusp forms

I've asked the background question here, which still left unanswered. Now I have a more precise question. In my homework I've been asked to prove that $$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...