On the Hecke algebra of a noncongruence subgroup

We begin by recalling standard facts concerning Hecke algebras and modular forms, for details of which the reader is referred to Chapter 3 of Shimura’s book [11]. By definition H (in Shimura’s notation, R(Γ, GL2(Q) )⊗Q) is the Q-algebra spanned by double cosets [ΓγΓ], for γ ∈ GL2(Q) with det γ > 0. Write as usual Mk(Γ) for the complex vector space of holomorphic modular forms of weight k ≥ 0, and Sk(Γ) for the subspace of cusp forms. There is a natural action of H on Mk(Γ), which preserves Sk(Γ). (In fact there is more than one way to normalise this action; the choice is irrelevant for this paper.)