I want to find all homomorphisms $ \phi: \mathbb{Z}_n \to \mathbb{C} \setminus \{0\}$. I have observed so far that $\mathbb{Z}_n$ is generated by $1$, i.e. $ \mathbb{Z}_n = \langle 1 \rangle$. I have also considered that $\mathbb{C} \setminus \{0\} = \langle 1 \rangle$.
So far, then I surmise that our homormophism $\phi$ must satisfy $\phi(1) = 1$.
We also observe that $n \cdot 1 = 0 \mod n$ for $\mathbb{Z}_n$. So this must mean that the homormophism $\phi$ must satisfy
$$\phi(n \cdot 1) = \phi(0) = \phi(n) \phi(1) $$
What other sorts of properties/relations must $\phi$ satisfy?
NOTE: I have read the other posts regarding "finding all homormorphisms between..." and they either deal with two finite groups or deal with symmetric/dihedral groups, so please do not link them in the comments.