Modular Forms with Only Nonnegative Coefficients

2025/07/23 by Jenkins, Paul, Rouse, Jeremy
#11F11 #11F30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.17949

Abstract

We study modular forms for \textrmSL₂(ℤ) with no negative Fourier coefficients. Let A(k) be the positive integer where if the first A(k) Fourier coefficients of a modular form of weight k for \textrmSL₂(ℤ) are nonnegative, then all of its Fourier coefficients are nonnegative, so that A(k) can be interpreted as a ``nonnegativity Sturm bound''. We give upper and lower bounds for A(k), as well as an upper bound on the nth Fourier coefficient of any form with no negative Fourier coefficients.

Citations

• Vanishing of Poincaré series for congruence subgroups
• The lean mathematical library

Related

• Non-vanishing Fourier coefficients of modular forms
• Quadratic minima and modular forms
• On Fourier coefficients of elliptic modular forms \bmod ℓ with applications to Siegel modular forms
• Large Fourier coefficients of half-integer weight modular forms
• Non-zero coefficients of half-integral weight modular forms mod ℓ