Group $\Gamma_1(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ (awaiting review)

The group $\Gamma_1(N)$ is the subgroup of $\textrm{SL}(2,\mathbb{Z})$ that consists of matrices that become upper triangular with ones on the diagonal when reduced modulo $N$. That is, \[ \Gamma_1(N) = \left\lbrace \left .\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in \textrm{SL}(2,\mathbb{Z})\ \right|\ a\equiv 1,\ c\equiv 0,\ d\equiv 1\pmod N \right\rbrace. \]