From Theorem 1.67 in [K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and \(q\)-series. CBMS Regional Conference Series in Mathematics 102. Providence, RI: American Mathematical Society (AMS). viii, 216p. (2004; Zbl 1119.11026)], it is known that every modular form for \(\text{SL}_2(\mathbb Z)\) is a linear combination of eta quotients. This follows from eta identities for the Eisenstein series \(E_4\) and \(E_6\). Ono asks for analogous results for other spaces of modular forms. With Dedekind’s eta function \[ \eta(z)= q^{1/24} \prod^\infty_{n=1} (1- q^n), \] \(q= e^{2\pi iz}\), an eta quotient of level \(N\) is a function of the form \[ f(z)= \prod_{d|N} \eta(dz)^{r(d)} \] with \(r(d)\in \mathbb Z\).
(In this paper it is not required that \(f\) should be holomorphic at the cusps.) The author proves that for any \(N\), every modular form of integral weight on \(\Gamma_0(N)\) is a rational function of eta quotients of levels dividing \(4N\). Moreover, he shows that every modular form on \(\Gamma_1(N)\), \(2\leq N\leq 4\), \(\Gamma_0(5)\), \(\Gamma_0(7)\), is a linear combination of eta quotients. For the proofs he presents specific identities for Eisenstein series on the respective groups.