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I’m computing Jacobi sum of cubic residue symbol as follows:

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
sum([K(a).residue_symbol(P,N)*K(1-a).residue_symbol(P,N) for a in [2..P.norm()-1] if a not in P and 1-a not in P])

The output is -3*z - 4. This is correct.

But I wrote another equivalent code:

def Jacobi(P,l,j):
x=sum([K(a).residue_symbol(P,l)^j*K(1-a).residue_symbol(P,l)^j for a in range(2, P.norm()-1) if a not in P and 1-a not in P])
return x

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
Jacobi(P,N,1)

Then the output is -4*z - 4, this is odd. Where is wrong?

I’m computing Jacobi sum of cubic residue symbol as follows:

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
sum([K(a).residue_symbol(P,N)*K(1-a).residue_symbol(P,N) for a in [2..P.norm()-1] if a not in P and 1-a not in P])

The output is -3*z - 4. This is correct.

But I wrote another equivalent code:

def Jacobi(P,l,j):
 x=sum([K(a).residue_symbol(P,l)^j*K(1-a).residue_symbol(P,l)^j for a in range(2, P.norm()-1) if a not in P and 1-a not in P])
 return x

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
Jacobi(P,N,1)

Then the output is -4*z - 4, this is odd. Where is wrong?

I’m computing Jacobi sum of cubic residue symbol as follows:

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
sum([K(a).residue_symbol(P,N)*K(1-a).residue_symbol(P,N) for a in [2..P.norm()-1] if a not in P and 1-a not in P])

The output is -3*z - 4. This is correct.

But I wrote another equivalent code:

def Jacobi(P,l,j):
    x=sum([K(a).residue_symbol(P,l)^j*K(1-a).residue_symbol(P,l)^j for a in range(2, P.norm()-1) if a not in P and 1-a not in P])
    return x

N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
Jacobi(P,N,1)

Then the output is -4*z - 4, this is odd. Where is wrong?