The trick is to write the $j$-invariant function in terms of Eisenstein series, whose $q$-expansions have a simple expression. See Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", Chapter 1, Section 7, and in particular, Proposition 7.4 and Remark 7.4.1.
Álvaro Lozano-Robledo
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