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A362878 Theta series of 18-dimensional lattice Kappa_18.

%I A362878 #16 May 13 2023 01:57:14
%S A362878 1,0,6480,157680,1596510,9488016,40681440,140492880,406046520,
%T A362878 1047312720,2426695200,5208293520,10421250750,19873356480,35716191840,
%U A362878 62355291696,104234541390,169488573120,267064691760,413777075760,619573504896,920235334320,1331744781600
%N A362878 Theta series of 18-dimensional lattice Kappa_18.
%C A362878 Theta series is an element of the space of modular forms on Gamma_1(3) with Kronecker character -3, weight 9, and dimension 4 over the integers.
%D A362878 J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
%H A362878 Andy Huchala, <a href="/A362878/b362878.txt">Table of n, a(n) for n = 0..20000</a>
%H A362878 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/KAPPA18.html">Home page for this lattice</a>.
%e A362878 G.f. = 1 + 6480*q^4 + 157680*q^6 + ...
%o A362878 (Magma)
%o A362878 prec := 20;
%o A362878 ls := [4,2,4,0,-2,4,0,-2,0,4,0,0,-2,0,4,-2,-2,0,0,0,4,-2,-1,1,0,0,0,4,-2,-1,0,-1,1,2,2,4,-2,-2,0,1,1,2,2,2,4,-2,0,-2,0,1,1,0,0,0,4,1,1,0,0,0,-2,0,-1,-1,-2,4,-2,-1,0,0,0,1,1,1,1,1,-2,4,0,-1,1,1,0,-1,1,0,0,-1,1,-1,4,0,0,0,0,0,0,0,0,0,0,0,0,-2,4,0,-1,0,0,1,1,0,1,1,-1,0,0,1,-1,4,0,0,1,0,-1,0,1,0,0,0,-1,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,0,1,4];
%o A362878 S := SymmetricMatrix(ls);
%o A362878 L := LatticeWithGram(S);
%o A362878 T<q> := ThetaSeries(L, 8);
%o A362878 M := ThetaSeriesModularFormSpace(L);
%o A362878 B := Basis(M, prec);
%o A362878 Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..4]]);
%Y A362878 Cf. A029897, A047628, A362875, A362876, A362877, A362879, A362880.
%K A362878 nonn
%O A362878 0,3
%A A362878 _Andy Huchala_, May 08 2023