Rational spanning sets of level-one cusp forms from Eisenstein series at prime levels

Abstract

Let $S_k$ be the space of cusp forms of weight $k$ for the full modular group ${\rm SL}_2(\mathbb{Z})$. In this paper, we will exhibit an explicit rational spanning subset of $S_k$ constructed from Eisenstein series at prime levels. The main ingredient of our proof is to show a result on determining cusp forms by noncentral quadratic-twist $L$-values.