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In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{t}$ of order $n_{t} = 2^{2^{t-1}-1}$ such that the minimal polynomial of $\\frac{1}{\\sqrt{n_{t}}}H_{t}$ is the cyclotomic polynomial $\\Phi_{2^{t+1}}(x)$. 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