ON THE MINUS PARTS OF CLASSICAL POINCAR´E SERIES

Abstract

Let Sk(N ) be the space of cusp forms of weight k for Γ0(N ). We show that Sk(N ) is the direct sum of subspaces S+ k (N ) and S− k (N ) is the vector space of cusp forms of weight k for the group Γ+ 0 (N ) generated by Γ0(N ) and WN and S− k (N ) is the subspace consisting of elements f in Sk(N ) satisfying f |kWN = −f . We find generators spanning the space S− k (N ) from Poincar´e series and give all linear relations among such generators.

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