Reference request for Hecke operators for principal congruence subgroup of modular group

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.

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.