While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is \begin{equation} \prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) \end{equation} where $\zeta$ is an arbitrary fifth of unity and \begin{equation} \varphi(q):=\sum_{n=-\infty}^{\infty}q^{n^2}=(q^2;q^2)_{\infty}(-q;q^2)_{\infty}^2 \qquad, |q|<1\\ (a;q)_{\infty}:= \prod_{n=0}^{\infty}(1-aq^n)\quad. \end{equation} Could anyone help me to understand how it holds?

While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is \begin{equation} \prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) \end{equation} where $\zeta$ is an arbitrary fifth root of unity and \begin{equation} \varphi(q):=\sum_{n=-\infty}^{\infty}q^{n^2}=(q^2;q^2)_{\infty}(-q;q^2)_{\infty}^2 \qquad, |q|<1\\ (a;q)_{\infty}:= \prod_{n=0}^{\infty}(1-aq^n)\quad. \end{equation} Could anyone help me to understand how it holds?