For odd primes p ≡ 21 or 29 mod 40, G_p(ζ) equals a sign given by the count of quadratic non-residues in a short interval, or that sign times ζ, where ζ is a primitive tenth root of unity.
Sun,Trigonometric identities and quadratic residues, Publ
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On a polynomial involving quadratic residues modulo primes
For odd primes p ≡ 21 or 29 mod 40, G_p(ζ) equals a sign given by the count of quadratic non-residues in a short interval, or that sign times ζ, where ζ is a primitive tenth root of unity.