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In this paper, the following result is proved. For any prime $p$, all the Fourier coefficients of $$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$ ...

$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by $$\Ord_p(F(q)):=\min\{...