It is known that( it is a theorem) every entire modular form of weight k is a polynomial in $G_4 $ and $ G_6$ of the type f = $\sum_{a, b} c_{a, b} G_{4}^a G_{6}^b $ where the $c_{a, b} $are complex numbers and the sum is extended over all integers a$\geq$0 and b$\geq$0 such that 4a+6b= k.
How can I prove that the products $G_{4}^a G_{6}^b $ are linearly independent where a, b are non negative integers satisfying 4a+6b= k?
Can someone please give hint.
$G_6(i)=0$, and $G_4(i)\ne0$. For $4a+6b=0$, the $G_4^aG_6^b$ have distinct orders of vanishing at $i$ and so are linearly independent.