On the parity of the Fourier coefficients of hauptmoduln $j_{N}(z)$ and $j_{N}^{+}(z)$

TL;DR

This paper investigates the parity properties of Fourier coefficients of hauptmoduln functions $j_N(z)$ and $j_N^+(z)$ for certain positive integers N, employing elementary methods and techniques inspired by Kolberg's proof.

Contribution

It provides new results on the parity of Fourier coefficients of hauptmoduln functions using elementary and classical techniques.

Findings

Identifies parity patterns in Fourier coefficients of specific hauptmoduln functions.

Extends Kolberg's parity proof techniques to modular functions.

Offers elementary proofs for parity results in modular forms.

Abstract

We obtain some interesting results about the parity of the Fourier coefficients of hauptmoduln $j_{N}(z)$ and $j_{N}^{+}(z),$ for some positive integers $N$. We use elementary methods and the techniques of O. Kolberg's proof for the parity of the partition function.