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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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up vote 6 down vote accepted

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.

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Thanks. I saw the explanation but couldn't link it to equation (2) as mentioned in the paper. –  Eugene Mar 13 '12 at 1:47
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