Vol. 9, No. 8, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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

Let f = n=0anqn be a modular form modulo a prime p, and let π(f,x) be the number of nonzero coefficients an for n < x. We give an asymptotic formula for π(f,x); namely, if f is not constant, then

π(f,x) c(f) x (logx)α(f)(loglogx)h(f),

where α(f) is a rational number such that 0 < α(f) 34, h(f) is a nonnegative integer and c(f) is a positive real number. We also discuss the equidistribution of the nonzero values of the coefficients an.

modular forms modulo $p$, Hecke operators, Selberg–Delange's method
Mathematical Subject Classification 2010
Primary: 11F33
Secondary: 11F25, 11N25, 11N37
Received: 29 October 2014
Revised: 5 July 2015
Accepted: 3 August 2015
Published: 29 October 2015
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