On the minus parts of classical Poincar\'e series

Abstract

Let $S_k(N)$ be the space of cusp forms of weight $k$ for $\Gamma_0(N)$. We show that $S_k(N)$ is the direct sum of subspaces $S_k^+(N)$ and $S_k^-(N)$. Where $S_k^+(N)$ is the vector space of cusp forms of weight $k$ for the group $\Gamma_0^+(N)$ generated by $\Gamma_0(N)$ and $W_N$ and $S_k^-(N)$ is the subspace consisting of elements $f$ in $S_k(N)$ satisfying $f|_kW_N=-f$. We find generators spanning the space $S_k^-(N)$ from Poincar\'e series and give all linear relations among such generators.