Which integer combinations of $n$-th roots of unity are zero?

Let $omega$ be primitive $n$-th root of unity. Can we determine all tuples of integers $(c_1, c_2,ldots,c_n) $ such that $$c_1+c_2 omega + c_3 omega^2+cdots+ c_n omega^{n-1}=0 ,?$$

It is clear to me that if $ n$ is prime, then this means $ omega$ is a root of polynomial $$c_1 + c_2x + c_3x^2+cdots+c_nx^{n-1} =0 ,,$$ which implies $c_1 = c_2 = c_3cdots = c_n $ as minimal polynomial of $omega$ in this case is $${1+x+x^2+dotsb+x^{n-1}},.$$ But if $ n$ is not prime and $phi(n) $ divides $(n-1) $ then other solutions are also possible. Does this become highly dependent on $ n$? Or can we still say something for general $ n$?