(A) numerical method to find an optimal independent system for certain congruence subgroups

Abstract

We implement the splitting algorithm based on brute force to construct a special polygon for $\Gamma_0(N)$ in the sense of Kulkarni and apply it to find an optimal value $m(\Gamma_0(N))$ denotes the minimum of the maximum value of the denominators among all possible special polygons. In the special case of $N=33$, we further determine whether the set of its special polygons contains a generalized Farey sequence with denominators less than or equal to $\lfloor \sqrt{N}\rfloor$ or not. Finally, we list up all types of special polygons for $\Gamma_0(N)$ when $N\leq 50$ is of the form $p$, $p^2$, or $pq$ for close odd primes $p$ and $q$.