For level 1 modular form, it is known that every modular form can be represented as a polynomial in weight 4 and 6 Eisenstein series. I wonder if this is true for higher levels. Interestingly, I found some papers that consider the problem of building modular forms as a linear combination of products of *at most two* of Eisenstein series. Here are related papers:

- Raum & Xia, All modular forms of weight 2 can be expressed by Eisenstein series
- Dickson & Neururer, PRODUCTS OF EISENSTEIN SERIES AND FOURIER EXPANSIONS OF MODULAR FORMS AT CUSPS

If we allow arbitrary number of products (of Eisenstein series), is it true that they generate all modular forms?