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I previously asked this question on mathoverflow but got no satisfactory answers so I'm posting it here as well:

So initially Dummit, Kisilevsky and McKay found all Dedekind $\eta$-products which are newforms. This result was subsequently generalized by Martin. Further, Ono and Martin provide an exhaustive list of modular elliptic curves whose associated modular forms is expressible as an $\eta$-quotient.

It is known that $$ E_4(z) = \dfrac{\eta(z)^{16}}{\eta(2z)^8} + 2^8 \cdot \dfrac{\eta(2z)^{16}}{\eta(z)^8} $$ and $$ E_6(z) = \dfrac{\eta(z)^{24}}{\eta(2z)^{12}} - 2^5 \cdot 3 \cdot 5 \cdot \eta(2z)^{12}- 2^9 \cdot 3 \cdot 11 \cdot \dfrac{\eta(2z)^{16}}{\eta(z)^8} + 2^{13} \cdot \dfrac{\eta(4z)^{24}}{\eta(12)^{12}}, $$ leading Ono to pose the question in his book "Web of Modularity", which spaces of modular forms are generated by eta-quotients. This question was addressed somewhat in this paper by Kilford.

My question is, what is the motivation for this question by Ono as most of these expressions are rather "messy". Is this just a question of interest in itself or are there further consequences to this open question? Why are spaces generated by $\eta$-quotients of interest as opposed to some other modular form such as the Eisenstein series?

Also is there an algorithm that given a modular form that can be expressed as a linear combination (or even rational function) of $\eta$-quotients, can compute the corresponding expression?

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