Degree and ramification index of a natural projection

Degree and ramification index of a natural projection

Let $\Gamma$ a subcongruence group of $\text{SL}_2(\mathbb{Z})$,
$\mathbb{H}^* = \mathbb{H} \cup \{\infty\} \cup \mathbb{Q}$ and
$\overline\Gamma = \Gamma / (\Gamma\cap\{\pm 1\})$. Given the natural
projection
$$\pi:\Gamma/\mathbb{H}^* \to {\text{SL}_2(\mathbb{Z})}/\mathbb{H}^*,$$
why does there hold for the degree of the mapping $\pi$:
$$\deg \pi = (\overline{\text{SL}_2(\mathbb{Z})}:\overline{\Gamma})$$
and for the ramification index:
$$e_y = (\overline{\text{SL}_2(\mathbb{Z})}_y:\overline{\Gamma}_y)$$
This is one exercise that has been "left to the reader". Unfortunately, by
starting from the definitions $\deg \pi = \sum_{y\in\pi^{-1}(x)} e_y$
where $\pi(z)=\sum_{n=e_y} a_n z^n, a_{e_y}\neq 0$, I failed to give a
proof.