  In this article we perform an extensive study of the spaces of automorphic
forms for GL(2) of weight two and level N, for N an ideal in the ring of
integers of the quartic CM field generated by the twelfth roots of unity. This
study is conducted through the computation of the group cohomology of the
congruence subgroup of upper triangular matrices modulo N, and the
corresponding Hecke action which this space admits. Combining this Hecke data
with the Faltings-Serre method for proving equivalence of Galois
representations, we are able to provide the first known examples of modular
elliptic curves over this field.
