\begin{align*}
\Gamma_0(N) := \left\{
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in SL_2(\mathbb{Z}) : c \equiv 0 \pmod N
\right\}.
\end{align*}
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Let $\Gamma_0(N)$ act on the upper half-plane $\mathcal{H} := \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$ by fractional linear transformations, and let