Subgroup $\Gamma_0(N)$ of $\textrm{SL}(2,\mathbb{Z})$ (reviewed)

The group $\Gamma_0(N)$ is the subgroup of $\textrm{SL}(2,\mathbb{Z})$ that consists of matrices whose lower-left entry is 0 modulo $N$. That is, \[ \Gamma_0(N) = \left\lbrace \left .\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \textrm{SL}(2,\mathbb{Z})\ \right|\ c\equiv 0 \bmod N \right\rbrace. \]