A027652 Values of Zagier's function J_1.

-1, 2, 0, 0, -248, 492, 0, 0, -4119, 7256, 0, 0, -33512, 53008, 0, 0, -192513, 287244, 0, 0, -885480, 1262512, 0, 0, -3493982, 4833456, 0, 0, -12288992, 16576512, 0, 0, -39493539, 52255768, 0, 0, -117966288, 153541020

Offset

-1,2

Comments

On page 2 of Zagier "Traces of Singular Moduli" he writes "On the other hand, we define a (meromorphic) modular form of weight 3/2 by the formula g(tau) = theta_1(tau)*E_4(4*tau)/eta(4*tau)^6 = q^{-1} - 2 + 248q^3 - 492q^4 + 4119q^7 - 7256q^8 + ..., (3)". - Michael Somos, Jul 04 2014

In Mathoverflow question 158075 Piezas writes "Zagier defines the modular form of weight 3/2, g(tau) = (eta^2(tau)/eta(2*tau))*(E_4(4*tau)/eta^6(4*tau)) = theta_4(tau)*eta^2(4*tau)*cbroot(j(4*tau)) which has the nice q-expansion (A027652, negated terms),". - Michael Somos, Jul 04 2014

References

M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..2000 from N. J. A. Sloane)

M. Kaneko, The Fourier coefficients and the singular moduli of the elliptic modular function j(tau), Memoirs Faculty Engin. Sci., Kyoto Inst. Technology, 44 (March 1996), pp. 1-5.

T. Piezas, Monstrous Moonshine for Thompson group Th?

D. Zagier, Traces of Singular Moduli

Example

G.f. = -1/q + 2 - 248*q^3 + 492*q^4 - 4119*q^7 + 7256*q^8 - 33512*q^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (-1/q) EllipticTheta[ 4, 0, q] QPochhammer[ q^4]^2 (QPochhammer[ q^4, q^8]^8 + 256 q^4 QPochhammer[ q^4, q^8]^-16), {q, 0, n}]; (* Michael Somos, Jul 19 2015 *)

Crossrefs

Cf. A134414, A027653, A027654, A027655.

Sequence in context: A183896 A228073 A306936 * A350022 A348264 A127282

Adjacent sequences: A027649 A027650 A027651 * A027653 A027654 A027655

Keyword

sign

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Jul 25 2006

Status

approved