  The Fricke-Macbeath surface is the unique Hurwitz surface of genus 7 with 504
conformal automorphisms. In this paper, we prove that the first eigenvalue of
the Laplacian on the Fricke-Macbeath surface has a sevenfold multiplicity and
contained in the interval $[1.23, 1.26]$. Further, we numerically identify the
7-dimensional representation of its automorphism group corresponding to the
eigenspace associated to the first eigenvalue. We also determine the Dirichlet
domain centered at 0 for a Fuchsian group that uniformizes the Fricke-Macbeath
surface, identifying the algebraic coordinates for its vertices.
