Optimal Farey sequence for the Congruence subgroup $\Gamma_0(2^{n})$

Abstract

We prove that $\Gamma_0(2^n)$ ($n\ge2$) 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 $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $\Gamma_0(2^n)$ whose side-pairings give a set of independent generators of $\Gamma_0(2^n)$.