Subspaces spanned by eigenforms with nonvanishing twisted central L -values

In this article, we construct explicit spanning sets for two spaces of modular forms. One is the subspace generated by integral-weight Hecke eigenforms with nonvanishing quadratic twisted central L -values. The other is a subspace generated by half-integral weight Hecke eigenforms with certain nonvanishing Fourier coefficients. Along the way, we show that these subspaces are isomorphic via the Shimura lift.