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We use this result to study the non-negativity of the coefficients of the unique modular form of weight $k$ with Fourier expansion \\[F_{k,0}(z) = 1 + O(q^{\\ell + 1}).\\] In particular, we show that $k = 81632$ is the largest weight for which all the coefficients of $F_{0,k}(z)$ are non-negative. 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