#### Vol. 9, No. 8, 2015

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

### Joël Bellaïche and Kannan Soundararajan

Vol. 9 (2015), No. 8, 1825–1856
DOI: 10.2140/ant.2015.9.1825
##### 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. 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 ${a}_{n}$.

##### Keywords
modular forms modulo $p$, Hecke operators, Selberg–Delange's method
##### Mathematical Subject Classification 2010
Primary: 11F33
Secondary: 11F25, 11N25, 11N37
##### Milestones
Received: 29 October 2014
Revised: 5 July 2015
Accepted: 3 August 2015
Published: 29 October 2015
##### Authors
 Joël Bellaïche Department of Mathematics Brandeis University 415 South Street Waltham, MA 02453 United States Kannan Soundararajan Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305-2125 United States