Bias of Root Numbers for Modular Newforms of Cubic Level

Abstract

Let $H_{2k}^{\pm }(N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2k$, 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 $\phi (N)$ 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.

12 pages; some changes made according to the comments of a referee; to appear in Proc. Amer. Math. Soc