Effective construction of Hilbert modular forms of half-integral weight

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.