The index of $\{\pm 1\}$ in $\gamma^{-1}\Gamma_0(N)\gamma$

Question

Let $N$ be a positive integer, and let $\Gamma_0(N)$ be the Hecke subgroup. Let $\gamma\in\mathrm{SL}_2(\mathbb{Z})$. My question is: what is the generator of

$$\gamma^{-1}\Gamma_0(N)\gamma/\{\pm 1\}\;\;?$$

I guess that is: $T^N$ with $T=\left(\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right)$.