For the circulant matrix $C$ of order $n=4$ with first row $[-1,1,1,1]$ say $$ C = Circ(-1,1,1,1) $$ we have the equality of vectors $$ [R(1),R(\omega),R(\omega^2),R(\omega^3)] = c [-1,1,1,1], $$ where $$ \omega=exp(2\pi i/4)=i, $$ $$ c =-2 $$ and $R(t)$ is the the representer polynomial of $C$ namely, $$ R(t)=-1+t+t^2+t^3. $$

Question: Are there other such matrices $C$ when $n =4k >4$ ?

More precisely: Let $k >1$ be an integer, and let $n=4k.$ We want a matrix $C$ such that

(a) $$ C = Circ(h_1, \ldots,h_n), $$ be a non-singular circulant matrix of order $n$ with $h_i \in \lbrace -1,1 \rbrace$ for all $i=1, \ldots,n.$

and

(b) For some nonzero integer $c \neq 0$ one has the equality of vectors $$ [R(1),R(\omega), \ldots, R(\omega^{n-1}] = c [h_1, \ldots,h_n]. $$ where $$ \omega=exp(2\pi i/n), $$ and $R(t)$ is the the representer polynomial of $C$ namely, $$ R(t)= h_1+h_2t + \cdots + h_{n}t^{n-1}. $$