  We determine the limiting distribution of the normalized Euler factors of an
abelian threefold A defined over a number field k when A is geometrically
isogenous to the cube of a CM elliptic curve defined over k. As an application,
we classify the Sato-Tate distributions of the Jacobians of twists of the
Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total.
We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting
distribution of the normalized Euler factors is not determined by the limiting
distributions of their coefficients.
