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A004675 Theta series of extremal even unimodular lattice in dimension 72.

%I A004675 #22 Oct 26 2023 01:00:37
%S A004675 1,0,0,0,6218175600,15281788354560,9026867482214400,
%T A004675 1989179450818560000,213006159759990870000,13144087517631410995200,
%U A004675 525100718690287495741440,14756609779472604266496000,310160311536865273422120000
%N A004675 Theta series of extremal even unimodular lattice in dimension 72.
%C A004675 The construction of such a lattice was announced by G. Nebe, Aug 12 2010. - _N. J. A. Sloane_, Aug 13 2010
%D A004675 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.
%H A004675 N. J. A. Sloane, <a href="/A004675/b004675.txt">Table of n, a(n) for n = 0..1000</a>
%H A004675 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 A004675 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 A004675 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 A004675 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 A004675 # get th2, th3, th4 = Jacobi theta constants out to degree maxd
%p A004675 maxd:=2001:
%p A004675 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 A004675 a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd):
%p A004675 th4:=series(subs(q=-q,th3),q,maxd):
%p A004675 # get Leech etc
%p A004675 t1:=th2^8+th3^8+th4^8: e8:=series(t1/2,q,maxd):
%p A004675 t1:=th2^8*th3^8*th4^8: delta24:=series(t1/256,q,maxd):
%p A004675 leech:=series(e8^3-720*delta24,q,maxd):
%p A004675 u1:=series(leech^3,q,maxd):
%p A004675 #u2:=series(leech^2*delta24,q,maxd):
%p A004675 u3:=series(leech*delta24^2,q,maxd):
%p A004675 u4:=series(delta24^3,q,maxd):
%p A004675 u5:=series(u1-589680*u3-78624000*u4,q,maxd);
%t A004675 terms = 13;
%t A004675 maxd = 2*terms;
%t A004675 th1 = EllipticTheta[1, 0, q];
%t A004675 th2 = EllipticTheta[2, 0, q];
%t A004675 th3 = EllipticTheta[3, 0, q];
%t A004675 th4 = th3 /. q -> -q;
%t A004675 t1 = th2^8 + th3^8 + th4^8;
%t A004675 e8 = Series[t1/2, {q, 0, maxd}];
%t A004675 t1 = th2^8*th3^8*th4^8;
%t A004675 delta24 = Series[t1/256, {q, 0, maxd}];
%t A004675 leech = Series[e8^3 - 720*delta24, {q, 0, maxd}];
%t A004675 u1 = Series[leech^3, {q, 0, maxd}];
%t A004675 u3 = Series[leech*delta24^2, {q, 0, maxd}];
%t A004675 u4 = Series[delta24^3, {q, 0, maxd}];
%t A004675 u5 = Series[u1 - 589680*u3 - 78624000*u4, {q, 0, maxd}];
%t A004675 CoefficientList[u5, q^2][[1 ;; terms]](* _Jean-François Alcover_, Jul 08 2017, adapted from Maple *)
%Y A004675 Cf. A018236.
%K A004675 nonn
%O A004675 0,5
%A A004675 _N. J. A. Sloane_