Bias of Root Numbers for Modular Newforms of Cubic Level

TL;DR

This paper derives explicit formulas for the difference in counts of modular newforms with positive and negative root numbers at cubic levels, revealing a bias towards root number +1 using an analytic approach.

Contribution

It introduces a root-number weighted Petersson formula and provides explicit formulas for root number bias in modular newforms of cubic level, a novel analytic result.

Findings

Explicit formulas for root number difference involving class numbers

Demonstrates a bias towards root number +1 in the distribution

Uses a new root-number weighted Petersson formula

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.