Root number bias for newforms

Martin, Kimball

doi:10.1090/proc/16463

Root number bias for newforms
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by Kimball Martin;

Proc. Amer. Math. Soc. 151 (2023), 3721-3736

DOI: https://doi.org/10.1090/proc/16463

Published electronically: June 16, 2023

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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.

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