= Solution
The map $X(\Gamma)\to X(\Gamma(1))\cong\mathbb P^1$ has degree $d$, the effective projective index. Ramification occurs only over the two elliptic points and the <modular cusp>. Over the order-two elliptic point, $r_2$ points have index one and the other $(d-r_2)/2$ points have index two, contributing $(d-r_2)/2$ to the <ramification divisor>. The analogous order-three contribution is $2(d-r_3)/3$.
The local degree at a <modular cusp> is its effective <cusp width>. The <cusp widths> sum to $d$, so the <modular cusp> contribution is $d-r_\infty$. The <Riemann-Hurwitz formula>, $2g(X)-2=d(2g(Y)-2)+\sum_P(e_P-1)$, now yields
$$
2g-2=-2d+\frac{d-r_2}{2}+\frac{2(d-r_3)}3+d-r_\infty.
$$
Therefore the <genus formula for a modular curve> is
$$
\boxed{g=1+\frac d{12}-\frac{r_2}4-\frac{r_3}3-\frac{r_\infty}2.}
$$
Using the index in $\mathrm{SL}_2$ without accounting for its center would give the wrong degree when $-I\notin\Gamma$.
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