We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions
5
5
and
q
−
4
q-4
over the finite field
F
q
\mathbb {F}_q
. Our main results are formulas for the coefficients of the quadratic residue weight enumerators for such codes. If
q
=
p
v
q=p^v
and we fix
v
v
and vary
p
p
, then our formulas for the coefficients of the dimension
q
−
4
q-4
code involve only polynomials in
p
p
and the trace of the
q
t
h
q^{\mathrm {th}}
and
(
q
/
p
2
)
th
(q/p^2)^{\text {th}}
Hecke operators acting on spaces of cusp forms for the congruence groups
SL
2
(
Z
)
,
Γ
0
(
2
)
\operatorname {SL}_2 (\mathbb {Z}), \Gamma _0(2)
, and
Γ
0
(
4
)
\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
2
-torsion over a fixed finite field.