Let \(\eta (z)\) be the Dedekind eta function. Newman [2, 3] studied the modularity of eta-quotients, giving necessary and sufficient conditions for a function of the form \(\prod _{0 < m \mid N} \eta (m z)^{r_m}\) to be a (weakly) holomorphic modular form of level N. We explain a proof of Newman’s theorem. The key observation is that although \(\Gamma _1(N)\) is not generated by \(\left( {\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}}\right) \) and \(\left( {\begin{smallmatrix} 1 & 0 \\ N & 1 \end{smallmatrix}}\right) \) , it is generated by those two matrices together with any congruence subgroup. Modularity with respect to some congruence subgroup is established using a simple identity involving the multiplier system of \(\eta (z)\) , whose proof is elementary in the sense that it avoids the use of Dedekind sums.
