Homomorphism on Abelian Group

Question

We are given that $G$ is an abelian group of order $n.$

If $f: G \rightarrow \mathbb{C}^*$ is any homomorphism, then show that $\sum_{g \in G} |f(g)| = n$

Please give a hint rather than the answer if possible.

Answer 1

Hint: Given an element $g\in G$, what values could $|f(g)|$ possibly take? (Note that that's absolute value, not algebraic order.)