Hecke fields

The Hecke field, or coefficient field, of a normalized Hecke eigenform is the number field generated by its Hecke eigenvalues. The degree of this field measures the size of its family of Galois-conjugate eigenforms.

Here this terminology is applied to the Bianchi and Hilbert newforms in the papers. See the LMFDB definition for classical newforms and its explanation of Hilbert Hecke orbits.