In page 3 of Kilford's paper generating spaces of modular forms with $\eta$-products, he mentions that there are only finitely many spaces of modular forms that can be completely generated by $\eta$-products. 


My question why is this true?


The paper can be found in following arXiv link:
<http://arxiv.org/pdf/math/0701478.pdf>


Thanks



Finitely many spaces generated by eta-products
There's a brief answer to this on page 3 of the paper; there are only finitely many eta-*products* with q-valuation 1 (one can write them all down), these all have a given level, so other levels will have a form with q-valuation 1 which can't be written as an eta-product. Note that I defined eta-products to be products of eta-functions with non-negative exponents.


