Sato-Tate groups in weight 0 and degree 1 are computed on demand.
Rational Sato-Tate groups in weight 1 and degree 2 are taken from Hyperelliptic curves, L-polynomials, and random matrices, by K.S. Kedlaya and A.V. Sutherland [arXiv:0803.4462, 10.1090/conm/487, MR:2555991].
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].
Rational Sato-Tate groups in weight 1 and degree 6 are taken from Sato-Tate groups of abelian threefolds: a preview of the classification, by F. Fité, K.S. Kedlaya, and A.V. Sutherland [arXiv:1911.02071] and Sato-Tate groups of abelian threefolds, by F. Fité, K.S. Kedlaya, and A.V. Sutherland [arXiv:2106.13759].
Additional acknowledgments
Data computed by Francesc Fité, Kiran Kedlaya, and Andrew Sutherland.