Let $E_2, E_4,$ and $E_6$ be the first three Eisenstein series. There are well-known formulas due to Ramanujan for the derivatives of these quantities:
\begin{align*} DE_2 &= \frac{1}{12}(E_2^2-E_4) \\ DE_4 &= \frac{1}{3}(E_2E_4-E_6) \\ DE_6 & = \frac{1}{2}(E_2E_6-E_4^2), \end{align*} where $(Df)(\tau):=\frac{1}{2\pi i}f'(\tau)$. A noticeable feature of these identities is that, for each $k=2,4,6$, there are constants $a_k, b_k$ such that $$DE_k - a_kE_2E_k = b_kE_{k+2}.\tag{*}$$ I was wondering if a formula of this type held for higher $k$. Now, I know that given a modular form $f\in\mathcal{M}_k$ with $k\geq 4$, the so-called Serre derivative $\vartheta f = Df - \frac{k}{12}E_2f$ has the property $\vartheta f\in\mathcal{M}_{k+2}$. If $\mathcal{M}_{k+2}$ happens to be one-dimensional, then a formula of the type (*) indeed holds with $a_k=k/12$. This means the minimal $k$ for which such a formula might fail is $k=10$ (so that $\mathcal{M}_{k+2} = \mathcal{M}_{12}$ is two-dimensional). In principle I know one can calculate a Fourier expansion for $\vartheta E_{10}$ and then write it in the basis of $\mathcal{M}_{12}$ by comparing coefficients, but I am having a hard time doing the calculation.
Even if (*) turns out to be false, I wonder if a similar-looking recurrence relation might still hold. Perhaps this is well-known result (I know very little about modular forms), but I couldn't find anything in my cursory glance of the literature.
So, I essentially have four questions:
- Does (*) hold for $k=10$?
- What about for higher $k$?
- Even if (*) is false, is there still some generalized recurrence formula on the derivatives?
- And if the answer to the previous question is "yes," where might I find the details?