Subgroup $\Gamma(N)$ of $\textrm{SL}(2,\mathbb{Z})$ (reviewed)

The group $\Gamma(N)$ is the subgroup of $\SL(2,\mathbb{Z})$ that consists of matrices congruent to the identity matrix modulo $N$. That is, \[ \Gamma(N)=\left\{ \begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\textrm{SL}(2,\mathbb{Z})\ \Bigg|\ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\equiv \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \bmod N \right\} . \]