  We determine the limiting distribution of the normalized Euler factors of an
abelian surface A defined over a number field k when A is isogenous to the
square of an elliptic curve defined over k with complex multiplication. As an
application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the
curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities
for the Sato-Tate group of an abelian surface defined over Q. With twists of
these two curves one encounters, in fact, all of the 18 possibilities for the
Sato-Tate group of an abelian surface that is isogenous to the square of an
elliptic curve with complex multiplication. Key to these results is the
twisting Sato-Tate group of a curve, which we introduce in order to study the
effect of twisting on the Sato-Tate group of its Jacobian.
