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

Baier, Harald; Köhler, Günter

Exp. Math. 12, No. 1, 115-121 (2003).

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 (This was based on earlier unpublished work of D. Zagier).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 \(j(sz)\).

Reviewer: Harvey Cohn (Laguna Woods)

MSC:

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

Keywords:

elliptic modular function; Fourier coefficients; Fourier series

Citations:

Zbl 1053.11042

References:

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