Character group of $\mathbb{Z}$

Question

I am trying to compute the character group of $\mathbb{Z}$ which contains homomorphisms that map into $\mathbb{C}^\times$. I have determined that each homomorphism $\phi \in \hat{\mathbb{Z}}$ may be written as $\phi(n) = \phi(1)^n$ for all $n \in \mathbb{Z}$. However, I am unsure whether there are any further restrictions I can pose on $\phi$ to properly determine $\hat{\mathbb{Z}}$. Any hints or comments are appreciated, thank you.