It is a classical result of Ribet that if an eigenform has CM the its residual projective image is "small" (cyclic or dihedral.) Is the converse true, i.e, if f is a form whose associated residual Gal representation has "small" projective image then f has CM? Thanks.



On the image of the residual representation attached to a CM form
No. For instance there are plenty of modular forms that are not CM, but are congruent mod p to CM forms or to Eisenstein series, and thus whose residual Galois representations have small image.


