A000521 Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau). (Formerly M5477 N2372)

1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075

Offset

-1,2

Comments

"The most natural normalization [of the j function] is to set the constant term equal to 24, the number given by Rademacher's infinite series for coefficients of the j function". [Borcherds]

Changing the term 744 to 24 gives A007240, the McKay-Thompson series of class 1A for Monster simple group.

sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

Klein's absolute invariant J=j/1728 is Gamma-modular.

(n+1)*A000521(n)/24 yields integral values - see A161395. - Alexander R. Povolotsky, Jun 09 2009

The Mathematica implementation of KleinInvariantJ[] (versions 6 to 8) had bugs giving wrong value for a[7], a[9], a[11] and other values. - Michael Somos, Mar 07 2012

It is an open question if there are infinitely many k such that a(k) is prime. The known such indices are listed in A339429. See the paper by Fredrik Johansson. - Peter Luschny, May 05 2021

References

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, pp. 376ff.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 20.

Evans, David E., and Yasuyuki Kawahigashi. "Subfactors and mathematical physics." Bulletin of the American Mathematical Society, 60:4, (2023), 459-482 (see page 472).

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.

M. J. Knopp, Rademacher on J(tau), Poincare series of nonpositive weights and Eichler cohomology, Notices Amer. Math. Soc., 37:4 (1990), 385-393.

S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.

J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from N. J. A. Sloane)

Hans-Fredrick Aas, Congruences for the Coefficients of the Modular Invariant j(tau), Mathematica Scandinavica, vol.15, pp. 64-68, 1964.

D. Alexander, C. Cummins, J. McKay, and C. Simons, Completely replicable functions, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165.

D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

H. Baier and G. Koehler, How to compute the coefficients of the elliptic modular function j(z), Experimental Mathematics 12 (2003).

R. E. Borcherds, Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006), Bull. Amer. Math. Soc., 45 (2008), 675-679.

J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

John Cremona, Home page

C. Daney, Open Questions:Elliptic Curves and Modular Forms

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.

Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.

Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067 [math.QA], 2001.

Y.-H. He and V. Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.

Yang-Hui He and John McKay, Moonshine and the Meaning of Life, arXiv:1408.2083 [math.NT], 2014.

Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.

M. Jankiewicz and T. W. Kephart, Transformations among large c conformal field theories, Nucl. Phys. B 744 (2006) 380-397 Table 6.

Fredrik Johansson, Computing isolated coefficients of the j-function, arXiv:2011.14671 [math.NT], 2020.

J. Jorgenson, L. Smajlovic, and H. Then, Kronecker's limit formula, holomorphic modular functions and q-expansions on certain moonshine groups, arXiv preprint arXiv:1309.0648 [math.NT], 2013.

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.

M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

K. Mahler, On a class of non-linear functional equations connected with modular functions, J. Austral. Math. Soc. Ser. A 22 (1976), no. 1, 65--118. MR0441867 (56 #258)

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Hisanori Mishima, Factorizations of many number sequences

William Stein, Database

Valdo Tatitscheff, A short introduction to Monstrous Moonshine, arXiv:1902.03118 [math.NT], 2019.

J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), 352-353.

Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.

Eric Weisstein's World of Mathematics, j-Function

Eric Weisstein's World of Mathematics, Monstrous Moonshine

A. van Wijngaarden, On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [ gives 100 terms ].

A. van Wijngaarden, On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [gives 100 terms]. [Annotated scanned copy]

Wikipedia, Modular lambda function

Herbert S. Zuckerman, The computation of the smaller coefficients of J(tau), Bull. Amer. Math. Soc. 45 (1939), 917-919.

Index entries for McKay-Thompson series for Monster simple group

Index entries for "core" sequences

Formula

G.f.: A007245(q)^3/q; or (1 + 240 Sum_{k>0} sigma_3(k) q^k )^3 / (q Product_{k>0} (1-q^k)^24 ).

It appears that -n * a(n) = A035230(n). - Gerald McGarvey, Dec 21 2006

2 * a(2) = A028520(3). 2 * a(4) + a(1) = A028520(4). 2 * a(6) = A028520(5). - Gerald McGarvey, Dec 21 2006

Expansion of 128 * (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) * (theta_2(q)^-8 + theta_3(q)^-8 + theta_4(q)^-8) in powers of q^2. - Michael Somos, Oct 02 2007

a(n) ~ exp(4*Pi*n^(1/2))/(2^(1/2)*n^(3/4)) [Petersson (1932), Rademacher (1938)]. - Gheorghe Coserea, Oct 09 2015

a(n) = (1/n)*(Sum_{r in Z} A027652(n - r^2) + Sum_{r>0, r odd} ((-1)^n * A027652(4*n - r^2) - A027652(16*n - r^2))) for n > 0. - Seiichi Manyama, Jun 11 2017

a(n) = (1/(n+1))*Sum_{k=1..n+1} (504*A001160(k) - 240*(n-k) * A001158(k)) * a(n-k), a(-1) = 1. - Seiichi Manyama, Jul 12 2017

G.f.: 256*(1 - lambda + lambda^2)^3/(lambda^2 * (1 - lambda)^2) where lambda is the elliptic modular function (A115977). - Seiichi Manyama, Jul 30 2017

Example

j = 1/q + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
From Seiichi Manyama, Jun 11 2017: (Start)
a(1) = (1/1)*(A027652(0) + A027652(1) + A027652(0) + (-A027652(3) - A027652(15) - A027652(7))) = (1/1) * 196884 = 196884.
a(2) = (1/2)*(A027652(1) + A027652(2) + A027652(1) + (A027652(7) + A027652(-1) - A027652(31) - A027652(23) - A027652(7))) = (1/2) * 42987520 = 21493760.
a(3) = (1/3)*(A027652(-1) + A027652(2) + A027652(3) + A027652(2) + A027652(-1) + (-A027652(11) - A027652(3) - A027652(47) - A027652(39) - A027652(23) - A027652(-1))) = (1/3) * 2592899910 = 864299970. (End)
If J_n := j(sqrt(-n))^(1/3), then J_1 = 12, J_2 = 20, J_4 = 66, J_77 = 255. - Michael Somos, Oct 31 2019

Maple

with(numtheory): TOP := 31;
g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));
g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));
delta := series(g2^3 - 27*g3^2, q, TOP);
j := series(1728 * g2^3 / delta, q, TOP);

Mathematica

CoefficientList[Normal[Series[1728*KleinInvariantJ[z], {z, 0, 30}]*Exp[ -2*I*Pi/z]] /. E^(Pi*Complex[0, n_]/z) -> t^(-n/2), t] (* Artur Jasinski, Dec 20 2008, after Daniel Lichtblau, corrected by Vaclav Kotesovec, Jul 07 2020 *)
a[ n_] := With[ {tau = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ 1728 KleinInvariantJ[ tau], {q, 0, n}], {q, 0, n}]]; (* Michael Somos, Nov 20 2011 *)(* Since V7 *)
a[ n_] := With[ {e1 = DedekindEta[ Log[q] / (2 Pi I)]^24, e2 = DedekindEta[ Log[q] / (Pi I)]^24}, SeriesCoefficient[ Series[ (e1 + 256 e2)^3 / (e1^2 e2), {q, 0, n + 1}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)
a[ n_] := With[ {L = ModularLambda[ Log[q] / (2 Pi I)]}, SeriesCoefficient[ Series[ 256 (L^2 - L + 1)^3 / (L (1 - L))^2, {q, 0, 2 n + 3}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)
a[ n_] := If[ n < -1, 0, With[ {E4 = 1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n + 2}], E6 = 1 - 504 Sum[ DivisorSigma[ 5, k] q^k, {k, n + 2}]}, SeriesCoefficient[ Series[ 1728 E4^3 / (E4^3 - E6^2), {q, 0, n}], {q, 0, n}]]]; (* Michael Somos, Mar 09 2012 *)
CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^3 / (16777216 * QPochhammer[-1, x]^24), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
a[n_] := SeriesCoefficient[With[{L = InverseEllipticNomeQ[rootQ]}, 256 (L^2 - L + 1)^3/(L (1 - L))^2], {rootQ, 0, 2n}];  (* Jan Mangaldan, Jul 07 2020, after Michael Somos; corrected by Leo C. Stein, Feb 25 2024 *)
a[n_] := SeriesCoefficient[ 12^3 KleinInvariantJ[Log[q]/(2 Pi I)], {q, 0, n}] (* Leo C. Stein, Feb 25 2024 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, A = x^(2*n + 2) * O(x); A = x * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8; polcoeff( subst( 256 * (1 - x + x^2)^3 / (x - x^2)^2, x, 16*A), 2*n))}; /* Michael Somos, Apr 30 2004 */
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^(5*n + 5) * O(x); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( subst( (x^2 + 10*x + 5)^3 / x, x, A), 5*n))}; /* Michael Somos, Apr 30 2004 */
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( (1 + 256*A)^3 / A, n))}; /* Michael Somos, Jul 13 2004 */
(PARI) q='q+O('q^66); Vec(ellj(q)) \\ Joerg Arndt, Apr 24 2016
(PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)), n))}; /* Michael Somos, Dec 25 2016 */

Crossrefs

Cf. A005798, A007240, A007245, A014708, A027652, A066395, A078906, A115977, A290403, A290404.

Reversion gives A091406 or A066396.

Cf. A106205 (24th root).

Cf. also A161361, A161362, A161395, A178451, A339429 (indices with prime values).

Sequence in context: A210178 A192731 A288261 * A178449 A178451 A066395

Adjacent sequences: A000518 A000519 A000520 * A000522 A000523 A000524

Keyword

easy,nonn,nice,core

Author

N. J. A. Sloane

Extensions

Expanded the definition to include additional search terms. - N. J. A. Sloane, Nov 30 2019

Status

approved