Let $M_k(\Gamma)$ be the integral weight $k$ modular forms defined over discrete subgroup $\Gamma\leq SL_2(R)$. One is given $dim_C M_k(\Gamma)\leq \frac{kV}{4\pi}+1$ with $V$ some constant number.
Book says the following. "If $f,g,h$ were algebraically independent modular forms, then for large $k$, dimension $M_k(\Gamma)$ would be at least the number of monomials in $f,g,h$ of total weight $k$, which is bigger than some positive multiple of $k^2$. This contradicts dimension of $M_k(\Gamma)$ growth in $k$ by above."
$\textbf{Q:}$ Suppose $f,g,h$ are weighted $l_1,l_2,l_3$ modular forms. I need to look for number of solutions asymptotic solution $(a_1,a_2,a_3)\in Z_{\geq 0}^3$ $a_1l_1+a_2l_2+a_3l_3=k$. It is clear that in general the defining equation cuts off a hyperplane in $R^3$ with a surface area asymptotic $k^2$. If I further assume the points are evenly distributed(very wrong hypothesis), then indeed I expect number of points lying on that hypersurface should be asymptotic to $k^2$ for large $k$. How do I argue that statement rigorously?
Ref. Zagier 1-2-3 Modular Forms pg 12. Proposition 3
In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(\Gamma)$ contains all the homogeneous polynomials of degree $k$ of $\mathbb{C}[f,g,h]$ which is a space of dimension $\frac{(k+2)(k+1)}{2}$.