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arXiv · 2507.17949

Modular Forms with Only Nonnegative Coefficients

Paul Jenkins·Jeremy Rouse
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Abstract

We study modular forms for $\textrm{SL}_2(\mathbb{Z})$ 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 $\textrm{SL}_2(\mathbb{Z})$ 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 $n$th Fourier coefficient of any form with no negative Fourier coefficients.

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Paul Jenkins, Jeremy Rouse. 2025-07-23. Modular Forms with Only Nonnegative Coefficients. https://arxiv.org/abs/2507.17949

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