  We study the quadratic residue weight enumerators of the dual projective
Reed-Solomon codes of dimensions $5$ and $q-4$ over the finite field
$\mathbb{F}_q$. Our main results are formulas for the coefficients of the the
quadratic residue weight enumerators for such codes. If $q=p^v$ and we fix $v$
and vary $p$ then our formulas for the coefficients of the dimension $q-4$ code
involve only polynomials in $p$ and the trace of the $q$th and $(q/p^2)$th
Hecke operators acting on spaces of cusp forms for the congruence groups
$\operatorname{SL}_2 (\mathbb{Z}), \Gamma_0(2)$, and $\Gamma_0(4)$. The main
tool we use is the Eichler-Selberg trace formula, which gives along the way a
variation of a theorem of Birch on the distribution of rational point counts
for elliptic curves with prescribed $2$-torsion over a fixed finite field.
