Ramanujan differential relations

For normalized Eisenstein series $E_2,E_4,E_6$ and $D=q\frac{d}{dq}$, the relations are $D{E}_2=(E_2^2-E_4)/12$, $D{E}_4=(E_2{E}_4-E_6)/3$, and $D{E}_6=(E_2{E}_6-E_4^2)/2$. They turn derivatives of these series into polynomial expressions in the same series.