p=7
G = DirichletGroup(p); G
m=3; n=ZZ((p-1)/m); print m,n
c=G[1]
c1=c^n;c1
The output is:
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!
EDIT: If I take the log and then exponentiate I get, for p=7;
$$ -\frac{3i}{2}\sqrt{3} +\frac{1}{2}$$
Not sure what this is exactly.
The question has been answered on Ask SageMath (by me) and also in the comments above.