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A362879 Theta series of 19-dimensional lattice Kappa_19.

%I A362879 #24 May 13 2023 01:57:04
%S A362879 1,0,9396,284528,3309660,21996036,103632480,384538752,1195104618,
%T A362879 3253783500,7971340896,17905302720,37530681590,74139276672,
%U A362879 139067432280,250102136592,433070833500,724358442744,1178016364548,1866143480400,2883345017508,4367172766500
%N A362879 Theta series of 19-dimensional lattice Kappa_19.
%C A362879 Theta series is an element of the space of modular forms on Gamma_0(12) of weight 19/2 and dimension 19 over the integers.
%D A362879 J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
%H A362879 Andy Huchala, <a href="/A362879/b362879.txt">Table of n, a(n) for n = 0..20000</a>
%H A362879 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/KAPPA19.html">Home page for this lattice</a>.
%e A362879 G.f. = 1 + 9396*q^4 + 284528*q^6 + ...
%o A362879 (Magma)
%o A362879 prec := 30;
%o A362879 coeffs := [1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548];
%o A362879 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, 1, 0, -1, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 4];
%o A362879 S := SymmetricMatrix(ls);
%o A362879 L := LatticeWithGram(S);
%o A362879 M := ThetaSeriesModularFormSpace(L);
%o A362879 B := Basis(M,prec);
%o A362879 Coefficients(&+[coeffs[i]*B[i] :i in [1..19]]);
%Y A362879 Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362880.
%K A362879 nonn
%O A362879 0,3
%A A362879 _Andy Huchala_, May 08 2023