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arXiv · 2008.08768

Bias of Root Numbers for Modular Newforms of Cubic Level

Qinghua Pi·Zhi Qi
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Abstract

Let $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $\varphi (N)$ and the class number of $\mathbb{Q}(\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.

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Qinghua Pi, Zhi Qi. 2020-08-20. Bias of Root Numbers for Modular Newforms of Cubic Level. https://arxiv.org/abs/2008.08768

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