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Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.
5

%I #33 Sep 08 2022 08:44:32

%S 1,0,0,26208,530712,6368544,47331648,256864608,1116087336,4092877152,

%T 12996075456,37058557536,96952754808,232778774592,526258264896,

%U 1128148021728,2286143305992,4451523096384,8386247967552,15130902687264,26614339616592,45684687301344

%N Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^12 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Dec 21 2015

%D N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

%H Vaclav Kotesovec, <a href="/A004046/b004046.txt">Table of n, a(n) for n = 0..2000</a>

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/L_24.2.html">Home page for lattice</a>

%H H.-G. Quebbemann, <a href="http://dx.doi.org/10.1006/jnth.1995.1111">Modular lattices in Euclidean spaces</a>, J. Number Theory, 54 (1995), 190-202.

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.

%F Theta series = a^12 - 9/2*a^8*b^4 + 414*a^6*b^6 + 1458*a^4*b^8 + 1998*a^2*b^10 + 459/2*b^12 (see PARI code for details).

%F G.f.: (27*a(x)^12 - 72*a(x)^9*b(x)^3 + 64*a(x)^6*b(x)^6 + 16*a(x)^3*b(x)^9 - 8*b(x)^12) / 27 where a(), b() are cubic AGM theta functions, - _Michael Somos_, Dec 25 2015

%e G.f. = 1 + 26208*x^3 + 530712*x^4 + 6368544*x^5 + 47331648*x^6 + ...

%e G.f. = 1 + 26208*q^6 + 530712*q^8 + 6368544*q^10 + 47331648*q^12 + ...

%t a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = ( 1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^4 (27 z^4 - 72 z^3 + 64 z^2 + 16 z - 8) / 27, {q, 0, n}]]]; (* _Michael Somos_, Dec 25 2015 *)

%o (PARI) th3 = sum(n=1,noo\2, 2*x^(4*n^2), 1+A);

%o th4 = sum(n=1,noo\2, (-1)^n*2*x^(4*n^2), 1+A);

%o th2 = sum(n=0,noo\2, 2*x^(4*n^2+4*n+1), A);

%o chk("th3^4 == th4^4+th2^4");

%o /* A004016(x^4) */

%o phi0 = th2*subst(th2,x,x^3)+ th3*subst(th3,x,x^3);

%o /* 2*x*A033762(x^2) */

%o phi1 = th2*subst(th3,x,x^3)+ th3*subst(th2,x,x^3);

%o /* A004010(x^2) */

%o K_12 = phi0^6+45*phi0^2*phi1^4+18*phi1^6;

%o a=phi0;b=phi1;

%o A004046=a^12-9/2*a^8*b^4+414*a^6*b^6+1458*a^4*b^8+1998*a^2*b^10+459/2*b^12;

%o (Magma) A := Basis( ModularForms( Gamma1(3), 12), 22); A[1] + 26208*A[4] + 530712*A[5]; /* _Michael Somos_, Dec 21 2015 */

%o (PARI) {a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^4 * (27*z^4 - 72*z^3 + 64*z^2 + 16*z - 8) / 27, n))}; /* _Michael Somos_, Dec 25 2015 */

%Y Cf. A107657.

%K nonn

%O 0,4

%A _N. J. A. Sloane_

%E PARI code from _Michael Somos_, Jun 07 2005