Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5

TL;DR

This paper derives explicit arithmetic formulas and asymptotic estimates for the Fourier coefficients of Hauptmoduln at levels 2, 3, and 5, extending previous results and employing advanced analytic techniques.

Contribution

It provides new explicit formulas and asymptotic behaviors for Hauptmoduln coefficients at specific levels, advancing understanding of their arithmetic properties.

Findings

Derived explicit arithmetic formulas for Fourier coefficients

Established asymptotic formulas using Murty-Sampath's method

Extended Kaneko's results to higher levels

Abstract

We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's result. We also obtain their asymptotic formulas by employing Murty-Sampath's method.