First $a_2$ moment (awaiting review)

The first $a_2$ moment of a Sato-Tate group $G$ is the expected value of the quadratic coefficient of the characteristic polynomial of a random element of $G$ (under its Haar measure).

When $G$ is the Sato-Tate group of an abelian variety, the first $a_2$ moment is equal to the rank of its Néron–Severi group [arXiv:1910.00518, MR:4038255].