For $n \geq 1$, I want to find all solutions $x_i$ of the equation

\begin{equation} \begin{array}l x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\ x_i^2 = 1, i=0,1,2\dotsc,n-1 \\ \omega = \cos(2\pi/n)+i\sin(2\pi/n) \\ z = \sum_{i=0}^{n-1} x_i \omega^{i} \\ \lvert z\rvert^2 \in \mathbb{Z}. \end{array} \end{equation}

As an example, $x_i = 1$, $i=0,1,2\dotsc,n-1$ is one solution to this equation. And $x_i = -1$, $i=0,1,2\dotsc,n-1$ is another solution. For small $n$, all solutions can be found by mathematical software. Is there any good idea for bigger $n$?

Here is the computational result for small $n$:

Here is the further detail, the prime number related patterns are quite obvious.

Actually, this problem has some variations, for example:

Consider $x_i = 1, 0$ or $x_i = \pm 1,0$ instead of $x_i = \pm 1$.

Consider remove the constraint $x_i^2=1$.

Consider $x_i \in \mathbb{Q}$ or $x_i \in \mathbb{R}$ instead of $x_i \in \mathbb{Z}$.

Any comment/answer to this problem and its variations will be appreciated.