Root-number bias conjecture for modular newforms of arbitrary level

Let $H^{+}_{2k}(M)$ and $H^{-}_{2k}(M)$ denote the subspaces of weight-$2k$ modular newforms of level $M$ having root number $+1$ and $-1$, respectively. Let $s_0^{\sharp}$ be the multiplicative function satisfying

$$s_0^{\sharp}(p)=1-\frac{1}{p},\qquad s_0^{\sharp}(p^2)=1-\frac{1}{p}-\frac{1}{p^2},\qquad s_0^{\sharp}(p^\alpha)=\left(1-\frac{1}{p}\right)\left(1-\frac{1}{p^2}\right)\quad(\alpha\geq 3).$$

Root-number bias conjecture. For any $M>1$,

$$|H^{+}_{2k}(M)|\geq |H^{-}_{2k}(M)|,$$

and the difference $|H^{+}_{2k}(M)|-|H^{-}_{2k}(M)|$ is independent of $k$ if $M$ is divisible by neither $2$ nor $3$ and $k>1$. Moreover,

$$|H^{\pm}_{2k}(M)|=\frac{2k-1}{24}M s_0^{\sharp}(M)+O(\sqrt{M}\log M).$$

This conjecture predicts a persistent bias toward positive root numbers and an asymptotic formula for the dimensions of the two root-number eigenspaces at arbitrary level. The preceding results establish the corresponding behavior for cubic levels, while the general assertion is left open.