Experimental Mathematics

Quadratic minima and modular forms

Barry Brent

Abstract

We give upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for $\flop{L}{-.3}\!_0(2)$. Numerical evidence indicates that a sharper bound holds for the weights $h \equiv 2 \pmod 4$. We derive upper bounds for the minimum positive integer represented by level-two even positive-definite quadratic forms. Our data suggest that, for certain meromorphic modular forms and $p=2$, $3$, the $p$-order of the constant term is related to the base-$p$ expansion of the order of the pole at infinity.

Article information

Source
Experiment. Math., Volume 7, Issue 3 (1998), 257-274.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674207

Mathematical Reviews number (MathSciNet)
MR1676754

Zentralblatt MATH identifier
0916.11025

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11E25: Sums of squares and representations by other particular quadratic forms

Keywords
Congruences constant terms Fourier series gaps modular forms quadratic forms quadratic minima

Citation

Brent, Barry. Quadratic minima and modular forms. Experiment. Math. 7 (1998), no. 3, 257--274. https://projecteuclid.org/euclid.em/1047674207


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