Modular Forms with Only Nonnegative Coefficients

TL;DR

This work investigates holomorphic modular forms for $SL_2(Z)$ whose Fourier coefficients are all nonnegative. It introduces $A(k)$ as a nonnegativity Sturm bound: if the first $A(k)$ coefficients are nonnegative, then all coefficients are nonnegative, and provides both lower and upper bounds for $A(k)$ as well as a bound on individual coefficients under the nonnegativity hypothesis. The authors prove a sharp-looking weight-asymptotic lower bound $A(k) > \frac{(k-1)^2}{16\pi^2}$ and an explicit upper bound $A(k) \le \frac{1}{7316} k^{4} (\log k + \log \log k)^{2}$ for $k \ge 12$, with $A(k)$ computed exactly for small weights up to $k=88$ and an algorithm handling $36 \le k \le 88$. The methodology combines Poincaré series analysis, the Eisenstein/cusp decomposition, Jenkins–Rouse bounds on cusp coefficients, and explicit linear-inequality searches to certify nonnegativity; the results illuminate sign-change phenomena in modular forms and have potential applications to theta series and related generating functions.

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.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark
• Theorem 4
• Theorem 5
• proof
• Lemma 6
• proof
• Lemma 7