Let $\Gamma$ be a congruence subgroup and let $F$ be its fundamental domain.
My first question is: Do the cusps of $\Gamma$ can be defined as elements of $\partial F\cap\mathbb{P^1(\mathbb{Q})}$? ($\partial F$ means the boundary of $F$).
My second question is: Does the fan width of $\Gamma_0(N)$ equal $1$.