Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated and let $n \geq 2$ be a natural number.

Question. Is there a way to compute $h^1(X, \, \operatorname{Sym}^n \Omega_X)$ in terms of the invariants of $X$ (Chern numbers, geometric genus, irregularity, etc.)? Or, at least, is it possible to compute it under some additional assumptions (either on $X$ or on $n$)?