Subgroup $\Gamma_1(N)$ of $\textrm{SL}(2,\mathbb{Z})$ (reviewed)

The group $\Gamma_1(N)$ is the subgroup of $\textrm{SL}(2,\mathbb{Z})$ defined by the congruences: \[ \Gamma_1(N) = \left\lbrace \left .\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \textrm{SL}(2,\mathbb{Z})\ \right| \ c\equiv 0,\ a \equiv d\equiv 1 \bmod N \right\rbrace. \]