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As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of $\\vartheta(\\tau,z)^4$, $\\eta(\\tau)^{12}$, $\\eta(\\tau)^4$ and $\\eta(\\tau)^8\\eta(2\\tau)^8$ in terms of Hurwitz class numbers, respectively. 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