A note on the spaces of Eisenstein series on general congruence subgroups

Abstract: This article proposes a new approach to study the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method provides a gateway to analytic and arithmetic properties of the spectral Eisenstein series using corresponding results for the principal congruence subgroup. We show that the specializations of the weight $k$ spectral Eisenstein series at $s = 0$ give rise to a basis for the space of Eisenstein series on a general congruence subgroup. A variant of this construction provides an algorithmically computable basis with algebraic Fourier coefficients for the same space. Our philosophy also yields an explicit basis parameterized by cusps for the space of Eisenstein series with nebentypus. We use this basis to generalize Hida's Eichler-Shimura isomorphism theorem for modular forms with nebentypus to almost all nebentypus characters and levels.