How to compute the coefficients of the elliptic modular function \(j(z)\). (English) Zbl 1053.11041

The authors consider various methods of computing all of the first \(N\) coefficients of \(j(z)\). Because of the recursion formulas, it is as easy to calculate the \(N\)th coefficient alone (with the memory effectively infinite). The numbers of multiplications are essentially all quadratic in \(N\) so the CPU running time is important for comparison. Particular attention is paid to the method of M. Kaneko [Traces of singular moduli and the Fourier coefficients of the elliptic modular function \(j(\tau)\), Fifth Conf. Canad. Number Theory Assoc. Ottawa, Ontario, Canada 1996, AMS, CRM Proc Lect. Notes 19, 173–176 (1999; Zbl 1053.11042)]. (This was based on earlier unpublished work of D. Zagier). The method involves the modular form of weight 3/2 namely \(g(z) = -E_4(4z)\theta_1(z)/\eta(4z)^6\), whose coefficients are related to those of \(j(sz)\).


11F30 Fourier coefficients of automorphic forms
11Y35 Analytic computations
11F03 Modular and automorphic functions
11F11 Holomorphic modular forms of integral weight


Zbl 1053.11042
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