Root number bias for newforms

TL;DR

This paper investigates the bias towards root number +1 in newforms across various levels, showing that the bias persists generally and identifying finite exceptions depending on the level's structure and weight.

Contribution

It extends the understanding of root number bias in newforms to arbitrary levels, characterizing when the bias holds and identifying finite exceptional levels.

Findings

Bias towards root number +1 persists for most levels.

Finite exceptional levels exist for fixed weight.

No exceptions for weights less than 12.

Abstract

Previously we observed that newforms obey a strict bias towards root number $+1$ in squarefree levels: at least half of the newforms in $S_k(\Gamma_0(N))$ with root number $+1$ for $N$ squarefree, and it is strictly more than half outside of a few special cases. Subsequently, other authors treated levels which are cubes of squarefree numbers. Here we treat arbitrary levels, and find that if the level is not the square of a squarefree number, this strict bias still holds for any weight. In fact the number of such exceptional levels is finite for fixed weight, and 0 if $k < 12$. We also investigate some variants of this question to better understand the exceptional levels.