Congruence classes for modular forms over small sets

Abstract

Serre showed that for any integer [Formula: see text] for almost all [Formula: see text] where [Formula: see text] is the [Formula: see text] Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study [Formula: see text] over the set of integers with [Formula: see text] many prime factors. Moreover, we show that any residue class [Formula: see text] can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of [Formula: see text]