The Dirichlet character modulo a prime number

Question

If I consider all Dirichlet characters modulo a prime number, let's say 5. Then I can have a form that looks like this:

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I found that

1. $$\begin{align*} \chi_{5,4}^2&= \chi_{5,1} \\ \chi_{5,3}^4&= \chi_{5,1} \\ &\vdots \end{align*}$$ That is, it seems that if I raise a character to some powers, I can get another one.

2. Also, the sum of each row is zero except for $\chi_{5,1}$, and the sum of each column is also zero except for $\chi_{5,4}$

Is this true for all Dirichlet characters modulo a prime number? If this is true, how should I prove these 2 facts? Thanks.

I'm a very beginner at number theory. If possible, please avoid using group theory and things relating to abstract algebra since I have no idea about what are they. Thank you very much!