Modular Forms with Only Nonnegative Coefficients

TL;DR

This paper investigates modular forms for SL2(Z) with nonnegative Fourier coefficients, defining a nonnegativity Sturm bound A(k), and provides bounds for A(k) and the coefficients.

Contribution

It introduces the concept of a nonnegativity Sturm bound A(k) for modular forms and establishes bounds for A(k) and the Fourier coefficients.

Findings

Derived upper and lower bounds for A(k).

Established an upper bound on the nth Fourier coefficient of forms with no negative coefficients.

Provided theoretical insights into the structure of nonnegative Fourier coefficient modular forms.

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.