%I #21 May 13 2023 01:58:00

%S 1,0,1746,21456,147150,607536,2036334,5410800,13282866,27563184,

%T 56679732,102040272,184563384,302221728,504866340,763016400,

%U 1202127174,1728479808,2575653198,3561176016,5127122304,6797385072,9531403128,12329627616,16701654486,21199654080

%N Theta series of 15-dimensional lattice Kappa_15.

%C Theta series is an element of the space of modular forms on Gamma_1(48) with Kronecker character 12 in modulus 48, weight 15/2, and dimension 58 over the integers.

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

%H Andy Huchala, <a href="/A362875/b362875.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/KAPPA15.1.html">Home page for this lattice</a>.

%e G.f. = 1 + 1746*q^4 + 21456*q^6 + 147150*q^8 + ...

%o (Magma)

%o prec := 70;

%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]);

%o ls := [1, 0, 1746, 21456, 147150, 607536, 2036334, 5410800, 13282866, 27563184, 56679732, 102040272, 184563384, 302221728, 504866340, 763016400, 1202127174, 1728479808, 2575653198, 3561176016, 5127122304, 6797385072, 9531403128, 12329627616, 16701654486, 21199654080, 28230179220, 34817427648, 45678519396, 55628679312, 71267532432, 85814825328, 108809427618, 128313065808, 161435864196, 188866349856, 233000967122, 271038881664, 332652360024, 380052936000, 464058384948, 528207272064, 634933480440, 719891109360, 862226645076, 963402396336, 1151630548200, 1283383148256, 1511712192624, 1682610190272, 1980149372586, 2173335020640, 2553938906832, 2802302452080, 3252053197962, 3565107859680, 4134281599332, 4478370612624];

%o L := LatticeWithGram(S);

%o M := ThetaSeriesModularFormSpace(L);

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

%o Coefficients(&+[ls[i] * B[i] : i in [1..58]]);

%Y Cf. A029897, A047628, A362876, A362877, A362878, A362879, A362880.

%K nonn

%O 0,3

%A _Andy Huchala_, May 07 2023