A362876 - OEIS

%I #17 May 10 2023 05:19:04

%S 1,0,2772,42624,335052,1545984,5698860,16297344,42785244,94440960,

%T 204094296,385391232,730053060,1240934400,2151268128,3374469504,

%U 5476016700,8115545088,12477938100,17677480320,26111897640,35570481408,50909418000,67336722432,93433877268

%N Theta series of 16-dimensional lattice Kappa_16.

%C Theta series is an element of the space of modular forms on Gamma_0(12) of weight 8 and dimension 17 over the integers.

%D J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

%H Andy Huchala, <a href="/A362876/b362876.txt">Table of n, a(n) for n = 0..20000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/KAPPA16.1.html">Home page for this lattice</a>.

%e G.f. = 1 + 2772*q^4 + 42624*q^6 + ...

%o (Magma)

%o prec := 40;

%o S := SymmetricMatrix([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,-1,4,0,0,0,0,0,0,1,0,1,-1,1,-1,1,4,0,0,0,0,-1,1,-1,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0,1,-1,1,0,0,-1,4]);

%o L := LatticeWithGram(S);

%o T<q> := ThetaSeries(L, 32);

%o M := ThetaSeriesModularFormSpace(L);

%o B := Basis(M,prec);

%o Coefficients(&+[Coefficients(T)[2*i-1]*B[i] : i in [1..17]]);

%Y Cf. A029897, A047628, A362875, A362877, A362878, A362879, A362880.

%K nonn

%O 0,3

%A _Andy Huchala_, May 07 2023