  A non-square-tiled Veech surface has finitely many periodic points, i.e.,
points with finite orbit under the affine automorphism group. We present an
algorithm that inputs a non-square-tiled Veech surface and outputs its set of
periodic points. We apply our algorithm to Prym eigenforms in the minimal
stratum in genus 3, proving that in low discriminant these surfaces do not have
periodic points, except for the fixed points of the Prym involution.
