In Brunier and Bundschuh, “On Borcherds Products Associated with Lattices of
Prime Discriminant.” Ramanujan Journal 7 (2003), 49–61, the authors use Borcherds lifts to obtain Hilbert modular forms.
Another approach is to calculate
Hilbert modular forms using the Jacquet--Langlands correspondence, which was implemented by Lassina Dembele
in "Magma". In Mayer, "Rings of Hilbert Modular Forms for the Fields $\Q(\sqrt{13})$ and $\Q(\sqrt{17})$,'' To appear, 2009,
we use Brunier and Bundschuh to determine the rings of Hilbert modular forms for
$\{Q}(\sqrt{13})$ and $\{Q}(\sqrt{17})$. In the present note we give the major
calculational details and present some results for
$\{K}=\{Q}(\sqrt{5})$, $\{K}=\{Q}(\sqrt{13})$, and $\K=\{Q}(\sqrt{17})$.
For calculations in the ring $\{o}$ of integers of $\{K}$ we order $\{o}$ by the norm of its
elements and get for fixed norm, modulo multiplication by $\pm \varepsilon_0^{2\Z}$, a finite set.
We use this decomposition to describe Weyl chambers and their boundaries, to determine the Weyl vector
of Borcherds products, and hence to calculate Borcherds products.
As a further example we calculate Fourier expansions of Eisenstein series.