login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
334
1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075 (list; graph; refs; listen; history; text; internal format)
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. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
John Cremona, Home page
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).
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.
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]
Herbert S. Zuckerman, The computation of the smaller coefficients of J(tau), Bull. Amer. Math. Soc. 45 (1939), 917-919.
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
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
KEYWORD
easy,nonn,nice,core
AUTHOR
EXTENSIONS
Expanded the definition to include additional search terms. - N. J. A. Sloane, Nov 30 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)