Fourth trace moment (awaiting review)

The fourth trace moment of a Sato-Tate group $G$ is the expected value of the fourth power of the trace of a random element of $G$ (under the Haar measure).

By a result known as "Larsen's alternative" [MR:2058618], if the fourth trace moment is less than or equal to 5 then $G$ is irreducible, and for $G$ of odd weight, the fourth trace moment is 3 when $G=\mathrm{USp}(2g)$ and greater than 3 otherwise.