Trace moments of a Sato-Tate group (reviewed)

The trace $t$ of a random element of a Sato-Tate group $G$ can be viewed as a random variable whose distribution is given by the pushforward of the Haar measure on $G$ under the trace map.

The $n$th moment $\mathrm{M}_n[t]:= \mathrm{E}[t^n]$ of $t$ is the expected value of the $n$th power of the trace, which is always an integer.

For Sato-Tate groups of odd weight we have $t=-a_1$ and the moment sequences of $t$ and $a_1$ coincide because their distributions are symmetric about zero.