Refined dimensions of cusp forms, and equidistribution and bias of signs

TL;DR

This paper refines the understanding of the structure and sign distributions of cusp forms of squarefree level, revealing biases in root numbers and Atkin-Lehner signs, with implications for Galois orbit counts.

Contribution

It provides refined dimension formulas for spaces of cusp forms with prescribed signs, extending previous results and uncovering biases in sign distributions.

Findings

Bias towards root number +1 in cusp forms

Sign patterns are biased in weight but equidistributed in level

Lower bounds on the number of Galois orbits

Abstract

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin-Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits.