I am stil stuck with the following:

Let $C$ be a symmetric circulant matrix with integer coefficients of order $n=4k$ (e.g., $C=circ(-1,1,1,1)).$

Assume that $C^{-1}$ is a polynomial (with rational coefficients) in $C;$ say $C^{-1} = P(C).$

Question: Can we get the `signature`

of the quadratic form $Q(x,y) = x^{T}Cy$ from this data ?

More precisely: How it may vary the signature depending on the polynomial $P,$ (if it happens that these signature depends at all on $P$).

In the case $C=circ(-1,1,1,1))$ the signature is $(1,3)$ that means: $1$ positive square and $3$ negative squares; moreover, we have $$ C^{-1} = C/4. $$