  Generically, one can attach to a Q-curve C octahedral representations
Gal(Qbar/Q) --> GL(2,Fbar_3) coming from the Galois action on the 3-torsion of
those abelian varieties of GL_2-type whose building block is C. When C is
defined over a quadratic field and has an isogeny of degree 2 to its Galois
conjugate, there exist such representations having image into GL(2,F_9). Going
the other way, we can ask which mod 3 octahedral representations of Gal(Qbar/Q)
arise from Q-curves in the above sense. We characterize those arising from
quadratic Q-curves of degree 2. The approach makes use of Galois embedding
techniques in GL(2,F_9), and the characterization can be given in terms of a
quartic polynomial defining the S_4-extension of Q attached to the octahedral
representation.
