Fourier coefficients of Hecke eigenforms

Abstract

We provide systematic evaluations, in terms of binary quadratic representations of $4p$, for the $p$-th Fourier coefficients of each member $f$ of an infinite class $\mathcal{C}$ of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share some properties with the eigenforms in $\mathcal{C}$, and we pose some conjectures about their Fourier coefficients.