Root number bias for newforms

Abstract

Previously we observed that newforms obey a strict bias towards root number + 1 +1 in squarefree levels: at least half of the newforms in S k ( Γ 0 ( N ) ) S_k(\Gamma _0(N)) with root number + 1 +1 for N 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 k > 12 . We also investigate some variants of this question to better understand the exceptional levels.