We give a method to
explicitly determine the space of unramified Hilbert cusp forms of weight
two, together with the action of Hecke, over a totally real number field of
even degree and narrow class number one. In particular, one can determine
the eigenforms in this space and compute their Hecke eigenvalues to any
reasonable degree. As an application we compute this space of cusp forms for
ℚ(), and determine each eigenform in this space which has rational Hecke
eigenvalues. We find that not all of these forms arise via base change from
classical forms. To each such eigenform f we attach an elliptic curve with
good reduction everywhere whose L-function agrees with that of f at every
place.
Keywords
Hilbert modular forms, elliptic curves, everywhere good
reduction
