An Explicit Theta Lift to Siegel Paramodular Forms

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an irreducible automorphic representation of $\mathrm{GSp}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.