We have the $j$-invariant defined as

I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.

The inversion formula for the $j$-invariant is $$ q=j^{-1}+\sum_{k\geq 2}d_kj^{-k}. $$

Thus, I would like to know some upper bound or asymptotic formula for $d_k$.

Any hint or reference?