Relationship between Dirichlet character and Legrendre symbol

Question

I'm wondering whether you always can express a non trivial Dirichlet character by a Legendre symbol. And in case so, how would one explicitly do that? Or how does one connect the two things?

Answer 1

Only $\pm 1$ valued Dirichlet characters are products of Legendre symbols $(n/p)$ (well almost all of them as there is a problem for the 3 quadratic characters $\bmod 8$)

If $\chi$ is a $\pm 1$ valued character of least period $2N+1$ then it is a product of $\pm 1$ valued characters of period $p^k \| 2N+1$ and each of them is either the trivial character $1_{p\ \nmid\ n}=(n/p)^2$ or the Legendre symbol $(n/p)$.

Then look at $$\chi(5n)=0,\chi(5n+1)=1,\chi(5n+2)=i,\chi(5n+3)=-i,\chi(5n+4)=-1$$