It is known that if
χ
\chi
is a real residue character modulo k with
χ
(
p
)
=
−
1
\chi (p) = - 1
for the first five primes p, then the corresponding Fekete polynomial
Σ
n
=
1
k
χ
(
n
)
x
n
\Sigma _{n = 1}^k\;\chi (n){x^n}
changes sign on (0, 1). In this paper it is shown that the condition that
χ
(
p
)
\chi (p)
be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if
χ
\chi
is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though
χ
(
2
)
=
χ
(
3
)
=
χ
(
5
)
=
χ
(
7
)
=
−
1
\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1
.