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 $k \geq 2$?



Sato-Tate conjecture for CM modular forms
Since the $L$-function of a CM modular form is just that of a Hecke character, the analogue of the Sato–Tate conjecture is much simpler to prove and follows from work of Deuring, I believe. If I remember correctly the measure one uses is (proportional to) $(1-z^2)^{-1/2}$ (and, as David points out, you only consider the primes that split in the associated imaginary quadratic field since $a\_p=0$ at all inert primes).


