TITLE: The cohomology of modular curves as a module over the Galois group
QUESTION [9 upvotes]: Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \mathbb C$ or in finite characteristic, on the etale cohomology group $H^1(X(N),\mathbb Z_\ell)\otimes_{\mathbb Z_\ell}\overline{\mathbb Q_\ell}$.
Do we know which irreducible representations of $G$ appear in the cohomology and with what multiplicities. Also, we can ask how the action of $G$ interacts with the Hecke operators, for instance. This seems to me to be very classical automorphic stuff but I have no knowledge about this area of math. Are there any friendly references?
Looking at the dimensions, I don't believe it is the regular representation.

REPLY [6 votes]: Jared Weinstein's PhD thesis (http://math.bu.edu/people/jsweinst/jswthesis.pdf) is an excellent reference for this kind of thing. See section 3.4 in particular, where he computes the space $S_k(\Gamma(N), \mathbb{C})$ as a $\mathbb{C}[\mathrm{SL}_2(\mathbb{Z}/N)]$-module using an equivariant version of the Riemann--Roch formula.