Shimura lifts of half-integral weight modular forms arising from theta functions

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.