Shimura lifts of certain classes of modular forms of half-integral weight

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map [Formula: see text]) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra’s work (on the image of a class of modular forms under the first Shimura map [Formula: see text]) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the [Formula: see text]th Shimura lifts [Formula: see text], for any positive square-free integer [Formula: see text].