A362875 Theta series of 15-dimensional lattice Kappa_15.

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

Offset: 0

Author

Andy Huchala, May 07 2023

Keywords

• nonn

Comments

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.

Examples

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

References

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

Links

• Andy Huchala, Table of n, a(n) for n = 0..20000
• G. Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

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

Programs

Magma

prec := 70;
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]);
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];
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M,prec);
Coefficients(&+[ls[i] * B[i] : i in [1..58]]);