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Let $\\mathcal{P}(n)$ be the set of monic irreducible polynomials of degree $n$ over $\\mathbb{F}_q$. For $f=t^n+f_{n-1}t^{n-1}+\\cdots+f_0\\in\\mathcal{P}(n)$, fix coefficients $c_0,\\ldots,c_m\\in\\mathbb{F}_q$ with $c_m\\ne0$ and put $$ Q_A(f)=\\sum_{j=0}^m c_j\\sum_{i=j}^n f_i f_{i-j}+\\ell_n(f),$$ where $\\ell_n$ is an arbitrary linear form in the coefficients of $f$ and $f_n=1$. 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