  We describe a construction of preimages for the Shimura map on Hilbert
modular forms, and give an explicit Waldspurger type formula relating their
Fourier coefficients to central values of twisted L-functions. Our construction
is inspired by that of Gross and applies to any nontrivial level and arbitrary
base field, subject to certain conditions on the Atkin-Lehner eigenvalues and
on the weight.
