Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\right)^n\right| = 1$ (note the complex modulus). Actually, I think that the only solutions are the trivial $k=an$, with $a \in \mathbb{Z}$. I know that similar problems can be really difficults, however I am not sure if this is the case.

Thank you very much for any suggestion. :-)