TITLE: SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N)
QUESTION [9 upvotes]: Since $\Gamma(N)$ is normal in $\mathrm{SL}(2,\mathbb{Z})$, the quotient group $\mathrm{SL}(2,\mathbb{Z}/N)$ acts on the spaces of cusp forms $S_k(\Gamma(N))$. How do these spaces decompose into irreducible representations? 
I can do the case $N=2$. I'm mostly interested in the case of $N$ a prime.

REPLY [7 votes]: As usual, once I spot a question on here I have anything useful to say about, somebody has already answered it.  
I can sum up that part of my thesis this way: let M be the induced representation of the character (-I) --> (-1)^k of the center up to all of SL(2,Z/NZ).  Then S_k(Gamma(N)) is roughly k/12 copies of M, plus some error term which can be given precisely, with some effort.  
When you ask instead about Hilbert modular forms over a totally real field K, the "1/12" becomes the absolute value of zeta_K(-1).