Unramified Hilbert modular forms, with examples relating to elliptic curves

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(root 509), 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.