Rational Sato-Tate groups in weight 1 and degree 4 are taken from Sato-Tate distributions and Galois endomorphism modules in genus $2$, by F. Fité, K.S. Kedlaya, V. Rotger, and A.V. Sutherland [arXiv:1110.6638 , 10.1112/S0010437X12000279 , MR:2982436 ].