|
%I #22 Oct 26 2023 01:00:37
%S 1,0,0,0,6218175600,15281788354560,9026867482214400,
%T 1989179450818560000,213006159759990870000,13144087517631410995200,
%U 525100718690287495741440,14756609779472604266496000,310160311536865273422120000
%N Theta series of extremal even unimodular lattice in dimension 72.
%C The construction of such a lattice was announced by G. Nebe, Aug 12 2010. - _N. J. A. Sloane_, Aug 13 2010
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.
%H N. J. A. Sloane, <a href="/A004675/b004675.txt">Table of n, a(n) for n = 0..1000</a>
%H J.-C. Belfiore and P. Sole, <a href="http://arxiv.org/abs/1010.4484">A Type II lattice of norm 8 in dimension 72</a>, arXiv:1010.4484 [cs.IT], 2010. - _N. J. A. Sloane_, Oct 23 2010
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/neb72.html">Home page for this lattice</a>
%H G. Nebe, <a href="http://arxiv.org/abs/1008.2862">An extremal even unimodular lattice of dimension 72</a>, Preprint, arXiv:1008.2862 [math.NT], Aug 12 2010. - _N. J. A. Sloane_, Aug 13 2010
%e Theta series begins 1 + 6218175600*q^8 + 15281788354560*q^10 + 9026867482214400*q^12 + 1989179450818560000*q^14 + 213006159759990870000*q^16 + 13144087517631410995200*q^18 + 525100718690287495741440*q^20 + 14756609779472604266496000*q^22 + ...
%p # get th2, th3, th4 = Jacobi theta constants out to degree maxd
%p maxd:=2001:
%p temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd):
%p a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd):
%p th4:=series(subs(q=-q,th3),q,maxd):
%p # get Leech etc
%p t1:=th2^8+th3^8+th4^8: e8:=series(t1/2,q,maxd):
%p t1:=th2^8*th3^8*th4^8: delta24:=series(t1/256,q,maxd):
%p leech:=series(e8^3-720*delta24,q,maxd):
%p u1:=series(leech^3,q,maxd):
%p #u2:=series(leech^2*delta24,q,maxd):
%p u3:=series(leech*delta24^2,q,maxd):
%p u4:=series(delta24^3,q,maxd):
%p u5:=series(u1-589680*u3-78624000*u4,q,maxd);
%t terms = 13;
%t maxd = 2*terms;
%t th1 = EllipticTheta[1, 0, q];
%t th2 = EllipticTheta[2, 0, q];
%t th3 = EllipticTheta[3, 0, q];
%t th4 = th3 /. q -> -q;
%t t1 = th2^8 + th3^8 + th4^8;
%t e8 = Series[t1/2, {q, 0, maxd}];
%t t1 = th2^8*th3^8*th4^8;
%t delta24 = Series[t1/256, {q, 0, maxd}];
%t leech = Series[e8^3 - 720*delta24, {q, 0, maxd}];
%t u1 = Series[leech^3, {q, 0, maxd}];
%t u3 = Series[leech*delta24^2, {q, 0, maxd}];
%t u4 = Series[delta24^3, {q, 0, maxd}];
%t u5 = Series[u1 - 589680*u3 - 78624000*u4, {q, 0, maxd}];
%t CoefficientList[u5, q^2][[1 ;; terms]](* _Jean-François Alcover_, Jul 08 2017, adapted from Maple *)
%Y Cf. A018236.
%K nonn
%O 0,5
%A _N. J. A. Sloane_
| 