Subspaces spanned by eigenforms with nonvanishing twisted central $L$-values

TL;DR

This paper constructs explicit bases for subspaces generated by eigenforms with nonvanishing twisted central $L$-values or specific Fourier coefficients, and shows their isomorphism via the Shimura lift.

Contribution

It provides explicit spanning sets for these subspaces and establishes their isomorphism through the Shimura lift, advancing understanding of eigenforms with special $L$-values.

Findings

Explicit spanning sets for the subspaces.

Isomorphism between the subspaces via Shimura lift.

Connection between nonvanishing $L$-values and Fourier coefficients.

Abstract

In this paper, we construct explicit spanning sets for two spaces. 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.