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

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