The database currently contains all rational Sato-Tate groups of weight 1 and degree up to 6, as well as the two rational Sato-Tate groups of weight 0 and degree 1. Pages for irrational Sato-Tate groups of weight 0 and degree 1 are available for component groups of order up to $10^{20}$, but only those order up to $10^{4}$ are shown in search results.
Completeness of Sato-Tate group data