It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the j-invariant is given by

$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\dots, \quad q = e^{2 \pi i \tau}.$$

I do not recall, however, ever seeing a modern treatment, nor even a hint, of how one might go about obtaining this expansion. Does anybody know a nice way to compute these coefficients? (I mean a way which does not invoke Moonshine, not that I'd expect that to make the computation more pleasant.) Is there a standard way to compute them?

I did find one approach published by H.S. Zuckerman in the late 1930s*, which makes use of a "fifth order multiplicator equation" for $j(\tau)$ -- distilled from Fricke and Klein's Vorlesungen uber die Theorie der elliptischen Modulfunktionen -- and an identity of Ramanujan for the generating function of partition numbers of the form $p(25n + 24)$. The treatment there is readable for the most part, but I don't know what a "multiplicator equation" is, and am having trouble seeing how he derived it from the pages he cites in Fricke-Klein.

*Zuckerman, Herbert S., The computation of the smaller coefficients of $J(\tau)$. Bull. Amer. Math. Soc. 45, (1939). 917–919.}

It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the j-invariant is given by

$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\dots, \quad q = e^{2 \pi i \tau}.$$

I do not recall, however, ever seeing a modern treatment, nor even a hint, of how one might go about obtaining this expansion. Does anybody know a nice way to compute these coefficients? (I mean a way which does not invoke Moonshine, not that I'd expect that to make the computation more pleasant.) Is there a standard way to do it?

I did find one approach published by H.S. Zuckerman in the late 1930s*, which makes use of a "fifth order multiplicator equation" for $j(\tau)$ -- distilled from Fricke and Klein's Vorlesungen uber die Theorie der elliptischen Modulfunktionen -- and an identity of Ramanujan for the generating function of partition numbers of the form $p(25n + 24)$. Is this typical?

*Zuckerman, Herbert S., The computation of the smaller coefficients of $J(\tau)$. Bull. Amer. Math. Soc. 45, (1939). 917–919.}