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More precisely, we consider the modular forms \\[F_{a,b,c}(z) := \\frac{\\eta(24az)^a \\eta(24acz)^{b-a}}{\\eta(24z)},\\] defined in terms of the Dedekind $\\eta$-function, for integers $a,c \\geq 1$ where $b \\geq 1$ is odd throughout. Serre determined the lacunarity of the series when $a = c = 1$. 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