The second Dirichlet coefficient starts out negative

TL;DR

This paper explores why classical modular forms of small weight and low level often have negative second Fourier coefficients, linking this phenomenon to properties of their associated L-functions.

Contribution

It provides an explanation for the observed negativity of the second Fourier coefficient in certain modular forms based on their L-functions.

Findings

Small weight, low level modular forms tend to have negative second Fourier coefficients.

Elliptic curve labeling schemes correlate with the rank of the curve.

L-functions explain the negativity phenomenon.

Abstract

Classical modular forms of small weight and low level are likely to have a negative second Fourier coefficient. Similarly, the labeling scheme for elliptic curves tends to give smaller labels to the higher-rank curves. These observations are easily made when browsing the L-functions and Modular Forms Database, available at http://www.LMFDB.org/. An explanation lies in the L-functions associated to these objects.