Klein's $j$-invariant (or the modular function $j$) appears in all introductory texts to modular forms but I have not seen a treatment of how to efficiently work out coefficients for the $q$-expansion of expressions like say, $j(\tau)^{-1/5}$ when the expansion of $j(\tau)$ is known.
One could just treat it by first principles and use the formula for the coefficients in a Fourier series, but that doesn't seem practical here. Is this discussed anywhere? These fractional powers of $j$ appear in the literature but have not seen a canonical explanation.