Every reference I've seen on modular forms seems to jump from the general definition of a modular form for congruence subgroups to studying modular forms just for $\Gamma\_0(N)$.


Once upon a time, I saw something like


$$M\_k(\Gamma(N)) \cong\bigoplus\_{\chi\mod N} M\_k(N^2,\chi)$$


where $M\_k(\Gamma(N))$ is the vector space of all weight $k$ modular forms for $\Gamma(N)$, and on the right side $\chi$ ranges over all dirichlet characters mod $N$, and $M\_k(N^2,\chi)$ is the vector space of weight $k$ modular forms $f$ satisfying
$$f(\gamma z) = \chi(d)(cz+d)^kf(z)$$
for all $\gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\in \Gamma\_0(N^2)$.


However, I don't see how this makes sense, since $\Gamma(N)$ is not a subgroup of $\Gamma\_0(N^2)$.



Why does it suffice to study modular forms for $\Gamma_0(N)$?
Conjugating $\Gamma(N)$ by $\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & N \end{bmatrix}$
 $$\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & N \end{bmatrix} \displaystyle\Gamma(N)\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & 1/N \end{bmatrix} 
\scriptstyle \ \ = \ \ \underbrace{\left\{\ \scriptstyle\begin{bmatrix} aN+1 & b \\ cN^2 & dN+1\end{bmatrix} \in SL\_2(\mathbb{Z})\right\}}\_{\displaystyle\tilde{\Gamma}\_1(N^2)}\displaystyle \ \supset \Gamma\_1(N^2)$$


Define the linear operator $T : M\_k(\Gamma(N)) \to M\_k(\tilde{\Gamma}\_1(N^2)), \ \ T f(\tau) = f (N\tau)$,


and the usual linear operators for showing $M\_k(\Gamma\_1(N^2)) =\displaystyle \bigoplus\_{\chi \bmod N^2}M\_k(\Gamma\_0(N^2),\chi)$ 


* For $gcd(d,N^2)=1$, let $\langle d \rangle : M\_k(\Gamma\_1(N^2)) \to M\_k(\Gamma\_1(N^2)) $, $\ \ \langle d \rangle g = g|\_k\gamma, \quad \gamma \in \Gamma\_0(N^2), \quad\gamma\_d \equiv d \bmod N^2$ (which is well-defined, not depending on the chosen $\gamma$). Note that $\langle d d' \rangle = \langle d \rangle\langle d' \rangle$
* And for a $\chi \bmod N^2$ :
$$ \pi\_\chi g= \frac{1}{\varphi(N^2)}\sum\_{\begin{array}{l}d \bmod N^2\\gcd(d,N^2) =1\end{array}} \overline{\chi(d) }\langle d \rangle g$$ 
$\pi\_\chi$ is an orthogonal projection $M\_k(\Gamma\_1(N^2)) \to M\_k(\Gamma\_0(N^2),\chi)$, and $\displaystyle\sum\_{\chi \bmod N^2} \pi\_\chi g=\langle 1\rangle g= g$ and for any $\chi \ne \chi'$ : $\pi\_\chi \pi\_{\chi'} = 0$
* Finally, $Tf = \langle dN+1\rangle T f$, so that $\langle dN+d'\rangle T f= \langle d'\rangle T f$ and hence $\pi\_\chi T f = 0$ whenever $\chi$ isn't a character $\bmod N$.


Thus $\sum\_{\chi \bmod N} \pi\_\chi Tf = Tf$, and together with $M\_k(\Gamma\_0(N^2),\chi) \subset M\_k(\tilde{\Gamma}\_1(N^2))$ for any $\chi \bmod N$, 


it means that $$M\_k(\Gamma(N)) \simeq M\_k(\tilde{\Gamma}\_1(N^2)) =\bigoplus\_{\chi \bmod N}M\_k(\Gamma\_0(N^2),\chi)$$


