Congruence classes for modular forms over small sets

Abstract

Serre showed that for any integer $m,a(n)\equiv 0(modm)$ for almost all $n,$ where $a(n)$ is the $n\text{th}$ Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study $\#{\{a(n)(modm)\}}_{n\le x}$ 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 13 Fourier coefficients, which are polynomially bounded as a function of $m.$