A362880 - OEIS

%I #15 May 13 2023 01:56:41

%S 1,0,15390,575160,7712820,57281580,296150580,1184012640,3944197800,

%T 11364334080,29395745478,69157229760,151652810580,311116423500,

%U 607158951120,1127694969072,2020055770530,3478103852940,5829999042420,9467119804680,15046034533560

%N Theta series of 20-dimensional lattice Kappa_20.

%C Theta series is an element of the space of modular forms on Gamma_0(9) of weight 10 and dimension 11 over the integers.

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

%H Andy Huchala, <a href="/A362880/b362880.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/KAPPA20.html">Home page for this lattice</a>.

%e G.f. = 1 + 15390*q^4 + 575160*q^6 + ...

%o (Magma)

%o prec := 40;

%o 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,-1,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,1,0,1,0,4];

%o S := SymmetricMatrix(ls);

%o L := LatticeWithGram(S);

%o M := ThetaSeriesModularFormSpace(L);

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

%o coeffs := [1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478];

%o Coefficients(&+[coeffs[i]*B[i] :i in [1..11]]);

%Y Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362879.

%K nonn

%O 0,3

%A _Andy Huchala_, May 08 2023