We know that $q$-expansion of the $j$-invariant function is given by $$ j(\tau)=q^{-1}+744+196884q+21493760q^2+\cdots = q^{-1}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{2\pi i\tau}$.

Someone know about the existence of an (almost) explicit inverse function of $j$? i.e., such that $$ q=j^{-1}+\sum_{k\geq 2}d_kj^{-k}, $$ where $j=j(\tau)$. I heard about such a formula and a few of initial values of $d_k$. In fact, $d_1=1$, $d_2=750420$, etc.

However, I don't know if there is some effective upper bound for $d_k$. For example, we know that $c_k\leq e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.

Any idea or reference where I can find something about that?