The number of nonzero coefficients of modular forms $(\mathrm{mod} p)$

Abstract

Let $f={\sum }_{n=0}^{\infty }{a}_n{q}^n$ be a modular form modulo a prime $p$, and let $\pi \left(f,x\right)$ be the number of nonzero coefficients $a_n$ for $n<x$. We give an asymptotic formula for $\pi \left(f,x\right)$; namely, if $f$ is not constant, then

$$\pi \left(f,x\right)\sim c\left(f\right)\frac{x}{{\left(logx\right)}^{\alpha \left(f\right)}}{\left(loglogx\right)}^{h\left(f\right)},$$

where $\alpha \left(f\right)$ is a rational number such that $0<\alpha \left(f\right)\le 3∕4$, $h\left(f\right)$ is a nonnegative integer and $c\left(f\right)$ is a positive real number. We also discuss the equidistribution of the nonzero values of the coefficients $a_n$.