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?