Root number bias for newforms
Kimball Martin
TL;DR
This work derives an explicit, largely class-number–driven formula for the trace of the Fricke involution on the new subspace $S_k^{\mathrm{new}}(N)$ of modular forms, yielding precise statements about the root-number bias among newforms of weight $k$ and level $N$. By combining Yamauchi–Skoruppa–Zagier trace formulas with elementary class-number relations and Möbius inversion, the author shows that the bias toward root number $+1$ persists for all weights when $N$ is not the square of a squarefree number (and in many small-weight cases), while for cubefree square levels the bias can flip for large $k$ and is controlled by the class-number data $h'(-D)$. The paper further refines these results into detailed, case-by-case formulas for $\operatorname{tr}W_N^{\mathrm{new}}$ and discusses how excluding twists from smaller levels (i.e., focusing on minimal newforms) can alter or neutralize the bias, highlighting rich interactions between local representations, twists, and global root-number statistics. Overall, the results give a complete, explicit understanding of root-number distributions in broad settings and connect root-number phenomena to classical arithmetic invariants and twist phenomena in the theory of modular forms.
Abstract
Previously we observed that newforms obey a strict bias towards root number $+1$ in squarefree levels: at least half of the newforms in $S_k(Γ_0(N))$ 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.