Root number correlation bias of Fourier coefficients of modular forms

Abstract:  In a recent paper, He, Lee, Oliver, and Pozdnyakov discovered a striking oscillating pattern in the average value of the P-th Dirichlet coefficients of elliptic curves for a prime P in a fixed conductor range with given rank. Later Sutherland performed computations that demonstrate that this bias extends to the P-th coefficients of modular newforms when correlated with the root number. I will prove this bias and compute the exact correlation function in the modular forms setting.