basis for $\mathcal M_{24}$

Question

When I run the command ModularForms(1, 24).basis() in Sage, I get back the following $q$-expansions:

$q + 195660q^3 + 12080128q^4 + 44656110q^5 + O(q^6)$

$q^2 - 48q^3 + 1080q^4 - 15040q^5 + O(q^6)$

$1 + \frac{131040}{236364091}q + \frac{1099243323360}{236364091}q^2 + \frac{12336522153621120}{236364091}q^3 + O(q^4)$

The second and third I recognize as $\Delta^2$ and $E_{24}$. However, I couldn't figure out how to express the first expansion in terms of $\Delta$ and $E_{2n}$. Is there a way to do this?

Answer 1

Got it after a little bit:

$$q + 195660q^3 + 12080128q^4 + 44656110q^5 + O(q^6) = \Delta(E_4^3 - 696\Delta).$$