Width of a cusp (reviewed)

The width of the cusp $\infty$ for the group $\Gamma$ is the smallest number $w$ such that $T^w=\left(\begin{matrix}1&w\\0&1\end{matrix}\right)\in\Gamma$. Furthermore, for a general $x\in\mathbb{P}^1(\mathbb{Q})$ and $\gamma\in\Gamma$ such that $\gamma\infty=x$, we define the width of $x$ for $\Gamma$ to be the width of $\infty$ for $\gamma^{-1}\Gamma\gamma$.

Note that $T=\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$ is one of the generators of the modular group $\textrm{SL}_2(\mathbb{Z})$.