Abstract
In this paper we describe the space of the cusp forms with the weight k for the congruence subgroup Γ 1( N), S k (Γ 1( N)), using Eichler-Shimura isomorphism, Shapiro lemma and the theory of group cohomology. An algorithm computing an integral basis of S k (Γ 1( N)) is developed. Applying the Hecke operators on the basis we can determine newforms in S k (Γ 1( N)).
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