Bias of Root Numbers for Modular Newforms of Cubic Level

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.