## Experimental Mathematics

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

### Quadratic minima and modular forms

#### 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