Applications and motivation of η-quotient generators and algorithms 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
$$
E_6(z) = \dfrac{\eta(z)^{24}}{\eta(2z)^{12}} - 2^5 \cdot 3 \cdot 5
\cdot \eta(2z)^{12} - 2^9 \cdot 3 \cdot 11 \cdot \dfrac{\eta(2z)^{12}\eta(4z)^8}{\eta(z)^8} + 2^{13} \cdot \dfrac{\eta(4z)^{24}}{\eta(2z)^{12}},
$$
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?
 A: The expressions are one of several possible expressions. The expression for $E_4(z)$ is about as nice as you can get since it is not an eta-quotient and thus requires at least two terms, while
$$ E_6(z) = (\eta(z)^8+2^5\eta(4z)^8)(\eta(z)^{16}-2^9\eta(z)^8\eta(4z)^8-2^{13}\eta(4z)^{16})/\eta(2z)^{12}$$
which is not that bad. The nice thing about eta-quotients is that they satisfy many algebraic identities and their transformation properties under the modular group is simple. Many modular forms can be expressed in terms of eta functions. Many more can be expressed in terms of products of $\;[q^k;q^n]:=(q^k;q^n)_\infty(q^{n-k};q^n)_\infty$ for fixed $n$ and $0<k<n,$ in addition to $(q^n;q^n)_\infty$. So now,
$$ f(z) = q\;[q;q^5]^2\;[q^2;q^5]^{-3}\;(q^5;q^5)^2_\infty = q-2q^2+4q^3-3q^4+q^5+\cdots,$$
where $q=e^{2\pi i z},$ is a modular form of weight $1$ for $\Gamma_1(5)$.
This may help answer your first question.
For your second question, using a computer algebra system that can handle power series, and a list of power series in one variable, it is possible to find a homogeneous (or not) polynomial relation between them of a given degree, if it exists, using essentially linear algebra to solve for the coefficients of the polynomial relation. As an example, the eta product identity of level 4 expressed in terms of $u_n:=(q^n)_\infty$ is
$\; 0 = u_1^{16}u_4^8 + 16qu_1^8u_4^{16} -u_2^{24}\;$ and this can be found using a program. More details of these identities is at my Dedekind Eta Function Product Identities website.
