#### 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