Let $A := \mathbb{C}[E_2;E_4,E_6]$ be the algebra of (almost) modular forms for $\mathrm{SL}_2(\mathbb{Z})$. (The weight $2$ Eisenstein series $E_2 := 1 - 24\sum_{n \geq 1} \sigma_1(n)q^n$ satisfies $E_2(-1/z) = z^2E_2(z) + 12z/2\pi i$ instead of $E_2(-1/z) = z^2E_2(z)$.) Ramanujan has shown that the derivation ("theta") operation $\Theta: \sum_{n \geq 0} a_nq^n \mapsto \sum_{n \geq 0} na_nq^n$ preserves the algebra $A$; more precisely, if $f$ is a weight $k$ modular form, then $\Theta f - \frac{k}{12}E_2f$ is a weight $k+2$ modular form.

On the other hand, since $E_2,E_4,E_6 \in \mathbb{Z}[[q]]$, the Fourier coefficients of any $f \in A$ are contained in a finitely generated free abelian group.

Question. Does the following converse to Ramanujan's result hold: If $f = \sum_{n \geq 1} a_nq^n \in A$ is a cusp form such that the $\mathbb{Z}$-module $\mathbb{Z}[a_n/n \mid n \geq 1]$ is finitely generated, then $\sum_{n \geq 1} \frac{a_n}{n} q^n \in A$ is again an (almost) modular cusp form?


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