Optimal Farey sequence for the Congruence subgroup ${\mathrm{\Gamma }}_0(2^n)$

We prove that ${\mathrm{\Gamma }}_0(2^n)$ ($n\ge 2$) has a Farey sequence $\{e_i\}$ such that $e_i\le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j=2^{n-1}$. For each $e_i$, there exists a unique $a_i$ such that $\{a_i\mathrm{/}{e}_i\}\cup \{\mathrm{\infty }\}$ is the set of ideal vertices of a fundamental domain of ${\mathrm{\Gamma }}_0(2^n)$ whose side-pairings give a set of independent generators of ${\mathrm{\Gamma }}_0(2^n)$.