Finitely many spaces generated by eta-products

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

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