Let $n>0$ be an even integer divisible by $4$

Let $R(t)=r_0+r_1t+ \cdots + r_{n-1}t^{n-1}$

be a polynomial with nonzero integer coefficients in $\{-1,1\}$ such that

$R(\omega)$ is a nonzero integer for all complex $\omega$ $\notin$ $\{-1,1\}$ such that $$ \omega^n=1 $$

Can we deduce that all these integers $R(\omega)$ have the same sign ???