Congruence classes for modular forms over small sets

TL;DR

This paper investigates the distribution of Fourier coefficients of certain cusp forms modulo m over integers with few prime factors, showing they can generate all residue classes with bounded sums.

Contribution

It introduces a class of cusp forms and demonstrates that any residue class modulo m can be expressed as a sum of at most thirteen Fourier coefficients, which are polynomially bounded.

Findings

Fourier coefficients of specific cusp forms can represent all residue classes mod m.

Any residue class mod m can be written as a sum of at most thirteen Fourier coefficients.

Fourier coefficients are polynomially bounded in m.

Abstract

J.P. Serre showed that for any integer $m,~a(n)\equiv 0 \pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cuspforms and study $\#\{a(n) \pmod m\}_{n\leq x}$ over the set of integers with $O(1)$ many prime factors. Moreover, we show that any residue class $a\in \mathbb{Z}/m\mathbb{Z}$ can be written as the sum of at most thirteen Fourier coefficients, which are polynomially bounded as a function of $m.$