How to make Dirichlet character table modulo $5$

Question

There are four reduced residue classes $\mod 5$, namely $1, 2, 3, 4$ and thus four Dirichlet characters $\mod 5$ since $\phi(5)=4$.

I understand how to deduce that the character can be $1$ or $0$(from defination):

$$\chi(n)=\begin{cases}1 &\text{ if } (n,5)=1\\0 &\text{ if } (n,5)>1\end{cases}$$

but i am not sure how to deduce for $i $ as shown here in $\mod 5$ table , using the following property we get:

$\chi(mn)=\chi(m)\chi(n)$

$\chi(4)=\chi(2)\chi(2)=i\times i=i^2=-1$

hence

$\chi_2(4)=\chi_4(4)=-1$

but,how is it known that

$\chi_2(2)=\chi_4(3)=i $

Answer 1

Every Dirichlet character $\chi$ (mod $5$) is determined by the value $\chi(2)$, since $(2,2^2,2^3,2^4) \equiv (2,4,3,1) \pmod 5$ and therefore $(\chi(1),\chi(2),\chi(3),\chi(4)) = (\chi(2)^4, \chi(2), \chi(2)^3, \chi(2)^2)$. (We're using the fact that $2$ is a "primitive root" modulo $5$.)

On the other hand, we already know that $\chi(1)=1$. Therefore $\chi(2)^4=1$, which means it's necessary that $\chi(2)$ is one of $\{i,-1,-i,1\}$.

This gives four possible Dirichlet characters, and you can check (or deduce from general theory) that all four possibilities really do yield Dirichlet characters.