On the vanishing of the coefficients of CM eta quotients

Abstract

This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $\eta(12z)^{2}$ and recent results of Chang on the pth coefficients of $\eta(4z)^{6}$ and $\eta(6z)^{4}$ . Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.