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Abstract
In [Brunier and Bundschuh 03], 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 09] we use [Brunier and Bundschuh 03]
to determine the rings of Hilbert modular forms for 
and 
. In the present note we give the major calculational
details and present some results for 
,
and 
. For calculations in the ring °
of integers of κ we order ° by the norm of its elements and get
for fixed norm, modulo multiplication by ±ϵ2ℤ
0, 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.