For a non-CM holomorphic modular forms of weight $k \geq 2$, the Sato–Tate conjecture is known to be true. Thanks to the work of Barnet-Lamb, Geraghty, Harris, and Taylor.

Do we have an analogous statement for CM modular forms as well? I mean, Is there a precise formulation (or a proof) of the Sato-Tate conjecture for CM modular forms of weight $k \geq 2$?