The Shimura lift and congruences for modular forms with the eta multiplier

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight $\lambda+1/2$ and level $N$ has weight $2\lambda$ and level $6N$, and is new at the primes $2$ and $3$ with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes $\ell$ which give rise to quadratic congruences modulo arbitrary powers of $\ell$.