Let $n \in \mathbb{N}$ and $i$ be the usual complex number such that $i^2=-1$.
Let $\zeta = \exp(\frac{\pi}{n} i)$, $v = [1,\zeta,\zeta^2,\ldots,\zeta^{n-1}]$. Given $J \subseteq \{1,\ldots,n\}$, let $u_J$ be the vector in $\mathbb{C}^n$ such that its $k^{th}$ entry is $-1$ if $k \in J$ and $1$ otherwise.
What is the dot product of $v$ and $u_J$?
I guess this is a relatively simple question and well understood in Number Theory (specifically cyclotomic fields?) and wondered if someone can point me to some citable references or useful information. This happens to have come up in a Coxeter Group context (projecting the vertices of a regular $n$-cube onto a Coxeter plane) and I would not be surprised if this is something Coxeter himself or someone with similar interests studied this.
Thanks in advance!