Congruence classes for modular forms over small sets

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 $m.$