Reference request for Hecke operators for principal congruence subgroup of modular group I am looking for references that discuss Hecke operators $T_n$ acting on modular forms
for the principal congruence subgroup $\Gamma(N)$ of the modular group $SL(2,Z)$ and am happy to restrict to the case that $(n,N)=1$. Most textbooks (Diamond and Shurman, Koblitz etc.) that discuss Hecke operators for congruence subgroups specialize to $\Gamma_0(N)$ or
$\Gamma_1(N)$ which contain $T: \tau \rightarrow \tau+1$, but I am specifically interested in the action of Hecke operators on modular forms which are invariant under $T^N$ but not under $T$ and are thus forms for $\Gamma(N)$ but not for $\Gamma_0(N)$ or $\Gamma_1(N)$. It seems reasonably clear how to obtain the answer using the double coset formulation of Hecke operators, but I have not found a reference which writes this out explicitly in terms of the relation between the coefficients of the Fourier expansion of the form at the cusp at infinity and the coefficients of its
image under $T_n$. 
 A: The Hecke operators $T(n)$ and the dual Hecke operators $T'(n)$ acting as correspondences on the modular curve $Y(N)$ are defined by Kato in $p$-adic Hodge theory and values of zeta functions of modular forms, section 2.9 (in Kato's notation $Y(N)=Y(N,N)$). The action of $T(p)$ on Fourier expansions is given in section 4.9, there he also describes the relation between his definition and other definitions in the litterature.
Actually, when you conjugate the double coset $\Gamma(N) \begin{pmatrix} n & 0 \\ 0 & 1 \end{pmatrix} \Gamma(N)$ in $\mathrm{GL}_2(\mathbf{Q})$ by the matrix $\begin{pmatrix} N & 0 \\ 0 & 1 \end{pmatrix}$, you get the double coset $\Gamma \begin{pmatrix} n & 0 \\ 0 & 1 \end{pmatrix} \Gamma$, where $\Gamma$ is this subgroup intermediate between $\Gamma_1(N^2)$ and $\Gamma_0(N^2)$. So it should be a simple exercise to check that Kato's definition agrees with David's answer.
A: The reason why Hecke theory for $\Gamma(N)$ doesn't get much treatment in the literature is because you can easily reduce it to the $\Gamma_1(N)$ case. More precisely, you can conjugate $\Gamma(N)$ by $\begin{pmatrix} N & 0 \\ 0 & 1\end{pmatrix}$ to get a group intermediate between $\Gamma_0(N^2)$ and $\Gamma_1(N^2)$.
This has come up before (in the context of explicit calcuations): see this question.
