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:

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

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    $\begingroup$ Khuri-Makdisi has proved that for every $N \geq 3$, the algebra generated by Eisenstein series of weight 1 on $\Gamma(N)$ contains all modular forms of weight $\geq 2$ on $\Gamma(N)$, see his article 'Moduli interpretation of Eisenstein series'. (Also, 'Neurural' should be 'Neururer'.) $\endgroup$ Jul 27, 2021 at 15:30
  • $\begingroup$ @FrançoisBrunault Thanks! I edited the name. $\endgroup$
    – Seewoo Lee
    Jul 28, 2021 at 3:40


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