LAUSR

Abstract Let Γ be a Fuchsian group of the first kind. The Eichler–Shimura isomorphism states that the space S k ( Γ ) is isomorphic to the first (parabolic) cohomology group associated to the Γ-module R k − 1 with an appropriate Γ-action. Manin reformulated the Eichler–Shimura isomorphism for the case Γ = SL 2 ( Z ) in terms of periods of cusp forms. In this paper we extend Manin's reformulation to the case Γ = Γ 0 + ( p ) with p ∈ { 2 , 3 } . The Manin relations describe relations between periods of cusp forms by using Hecke operators and continued fractions. We also extend the Manin relations and homogeneity theorem to cusp forms on Γ 0 + ( 2 ) without using continued fractions.