Suppose we have a function $\phi\colon \mathfrak H \longrightarrow \mathbb C$ such that


1. $\phi^{24}$ is a modular function of level $5$.
2. $\phi(\tau)=\sum\_{n=-1}^{\infty}a\_{n}q^{n/5}$, $a\_{-1}\neq 0,q=e^{2\pi i\tau}$.


Does it follow that $\phi$ is a modular function of level $5$?


In particular, I am interested in the function
$$\phi(\tau)=-\frac{1}{5^{1/2}}\frac{\eta(\tfrac{\tau}{5})\eta(\tfrac{\tau+1}{5})\eta(\tfrac{\tau-1}{5})}{\eta(5\tau)\eta(\tfrac{\tau+2}{5})\eta(\tfrac{\tau-2}{5})}.$$
Is there a a simple way to show that it is modular of level $5$? [Siegel](https://link.springer.com/article/10.1007/BF01425549) says that it is easy to show that it is so. We could compute the generators for $\Gamma(5)$ and then use the transformation formula for the Dedekind eta function, but this seems like a mess.



Why is this function a modular function of level $5$?
Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.


**Claim**: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is a modular function for $\Gamma\_{0}(25) \cap \Gamma\_{1}(5)$. (This is straightforward to prove using properties of the slash operator.)


Using this claim, it's not hard to see that $\eta(\tau/5)/\eta(5\tau)$ is a modular function of level $5$, using the standard result of Gordon, Hughes and Newman about when an eta quotient is modular of level $N$. (This result can be found in the [Wikipedia article about the Dedekind $\eta$-function](https://en.wikipedia.org/wiki/Dedekind_eta_function).)


It remains to show that $\frac{\eta(\frac{\tau+1}{5})\eta(\frac{\tau-1}{5})}{\eta(\frac{\tau+2}{5})\eta(\frac{\tau-2}{5})}$ is a modular function of level $5$.


The Siegel functions $g\_{(a\_{1}/N,a\_{2}/N)}$ can be used to build modular functions of level $N$. (See Section 5 of the article [here](https://afolsom.people.amherst.edu/Folsom-JNTB-publ.pdf) by Amanda Folsom that gives a product expansion of the Siegel functions and criteria of Kubert and Lang that indicate when a product of Siegel functions is modular.) The product formula implies that
$$
 g\_{(0,a\_{2}/N)}(\tau) = c \eta(\tau + a\_{2}/N) \eta(\tau - a\_{2}/N)
$$
for some constant $c$. The modularity criteria imply that
$$
 h(\tau) = \frac{g\_{(0,1/5)}(\tau)}{g\_{(0,2/5)}(\tau)} = \frac{\eta(\tau + 1/5) \eta(\tau - 1/5)}{\eta(\tau + 2/5) \eta(\tau - 2/5)}
$$
is a modular function for $\Gamma(5)$. Now $h(\tau) = h(\tau + 1)$ and so $h(\tau)$ is a modular function for $\left\langle \Gamma(5), \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\right\rangle = \Gamma\_{1}(5)$. Since $\Gamma\_{0}(25) \cap \Gamma\_{1}(5) \subset \Gamma\_{1}(5)$, the claim above implies that $h(\tau/5)$ is a modular function for $\Gamma(5)$ and this proves that $\phi$ is a modular function of level $5$.


As a note related to your first question about $\phi^{24}$ being modular of level $5$ implying $\phi$ is modular of level $5$, Theorem 1 of Kubert and Lang's paper [here](https://link.springer.com/content/pdf/10.1007/BF01361857.pdf) is the following. Let $N$ be a prime power and $U\_{N}$ be the group of modular units of level $N$ that are generated by the Siegel functions. If $g$ is a modular function and there is a positive integer $k$ so that $g^{k} \in U\_{N}$, then $g \in U\_{N}$.


