Suppose *f* is a newform (with coefficients generating some number field E), and $\rho_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}_2(E_\lambda)$ the associated Galois rep (for some prime $\lambda$ of E). When does $\rho$ have open image in ${\rm GL}_2(E_\lambda)$?

This clearly isn't the case if *f* has weight 1, or if *f* is of CM type; and I gather that it's a theorem of Serre that if *f* is attached to an elliptic curve, then not having CM is sufficient. What's known about this question in general?

cognoscenti? $\endgroup$ – Pete L. Clark May 10 '10 at 15:56