Experimental Mathematics

Quadratic minima and modular forms

Barry Brent


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

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

First available in Project Euclid: 14 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

Export citation