3614.sagews

D = DirichletGroup(13)

e = D[6]
f = D[3]

#Notice that J(e,f)=J(f,e).
#This Jacobi sum corresponds to the fact that we can write 13 as the sum of two squares:  13=4+9.
#zeta12^3=i

e.jacobi_sum(f)

2*zeta12^3 - 3

f.jacobi_sum(e)

2*zeta12^3 - 3

D = DirichletGroup(29)

e=D[14]  #This is the Legendre symbol
f=D[4]   #This gives fourth powers of zeta28; that is, it gives us 7th roots of unity.
g=D[8]
zeta28=CC.zeta(28)

#This shows that (1 + 2zeta7 + 2zeta7^2 + 2zeta7^4) = sqrt(-7).
(1+2*CC.zeta(7)+2*CC.zeta(7)^2+2*CC.zeta(7)^4)^2

-7.00000000000000 - 2.34989921838088e-15*I

e.jacobi_sum(f)

-2*zeta28^10 + 2*zeta28^4 + 4*zeta28^2 - 1

-2*zeta28^10 + 2*zeta28^4 + 4*zeta28^2 - 1

5.09783467904461 + 1.73553495647023*I

g.jacobi_sum(f)

4*zeta28^8 + 4*zeta28^4 - 4*zeta28^2 + 1

4*zeta28^8 + 4*zeta28^4 - 4*zeta28^2 + 1

-0.999999999999999 + 5.29150262212918*I

D=DirichletGroup(127)

e=D[18]
f=D[36]
zeta126=CC.zeta(126)

e.jacobi_sum(f)

-6*zeta126^33 + 6*zeta126^27 + 6*zeta126^24 + 6*zeta126^12 - 6*zeta126^3 - 11

-6*zeta126^33 + 6*zeta126^27 + 6*zeta126^24 + 6*zeta126^12 - 6*zeta126^3 - 11

-8.00000000000001 + 7.93725393319377*I