Primitive character, coductor and Jacobi sum

Dirichlet characters

Recall that is a character \chi of modulo q is said to be induced by a character \chi' of modulo d if \chi(n)=\chi'(n) for every n\in\mathbf{Z} with \gcd(n,q)=1, here d is a divisor of q.

Jacobi sum

Jacobi sum is a type of character sum formed with Dirichlet characters. The Jacobi sms for Dirichlet characters \chi,\chi' modulo a prime number p, defined by

\displaystyle J(\chi,\chi')=\sum_{n\in \mathbf{Z}/p\mathbf{Z}} \chi(n)\chi(1-n),

Jacobi sums are the analogues for finite fields of the beta function.