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 (This was based on earlier unpublished work of D. Zagier).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

### MSC:

 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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### References:

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