Chebyshev's bias for modular forms

Abstract

We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $\Gamma_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples.