Equations of Riemann surfaces with automorphisms

We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large automorphism groups (that is, $\mathrm{\mid }\mathrm{A}\mathrm{u}\mathrm{t}(X)\mathrm{\mid }>4(g_X-1)$) for genus $4\le g_X\le 7$. The main tools are the Eichler trace formula for the character of the action of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ on holomorphic differentials, algorithms for producing matrix generators of a representation of a finite group with a specified irreducible character, and Gr"obner basis techniques for computing flattening stratifications.