  We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed
field $k$ of characteristic $p\geq0$ and study the action of the automorphism
group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes
S_3$ on the canonical ring $R=\bigoplus H^0(F_n,\Omega_{F_n}^{\otimes m})$ when
$p>3$, $p\nmid n$ and $n-1$ is not a power of $p$. In particular, we explicitly
determine the classes $[H^0(F_n,\Omega_{F_n}^{\otimes m})]$ in the Grothendieck
group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective
equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our
results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary
Fermat curve.
