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ubuntu2004
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def gauss_sum(a,p):
    K.<zeta> = CyclotomicField(p)
    return sum(legendre_symbol(n,p) * zeta^(a*n) for n in range(p))

g1=gauss_sum(1,5); g1
-2*zeta^3 - 2*zeta^2 - 1 
g2=gauss_sum(2,5); g2
2*zeta^3 + 2*zeta^2 + 1 
g3=gauss_sum(3,5); g3
g4=gauss_sum(4,5); g4
g5=gauss_sum(5,5); g5
2*zeta^3 + 2*zeta^2 + 1 -2*zeta^3 - 2*zeta^2 - 1 0 
g1=gauss_sum(1,7); g1
2*zeta^4 + 2*zeta^2 + 2*zeta + 1 
g2=gauss_sum(2,7); g2
g3=gauss_sum(3,7); g3
g4=gauss_sum(4,7); g4
g5=gauss_sum(5,7); g5
g6=gauss_sum(6,7); g6
g7=gauss_sum(7,7); g7
2*zeta^4 + 2*zeta^2 + 2*zeta + 1 -2*zeta^4 - 2*zeta^2 - 2*zeta - 1 2*zeta^4 + 2*zeta^2 + 2*zeta + 1 -2*zeta^4 - 2*zeta^2 - 2*zeta - 1 -2*zeta^4 - 2*zeta^2 - 2*zeta - 1 0