A362878 Theta series of 18-dimensional lattice Kappa_18.

1, 0, 6480, 157680, 1596510, 9488016, 40681440, 140492880, 406046520, 1047312720, 2426695200, 5208293520, 10421250750, 19873356480, 35716191840, 62355291696, 104234541390, 169488573120, 267064691760, 413777075760, 619573504896, 920235334320, 1331744781600

Offset: 0

Author

Andy Huchala, May 08 2023

Keywords

• nonn

Comments

Theta series is an element of the space of modular forms on Gamma_1(3) with Kronecker character -3, weight 9, and dimension 4 over the integers.

Examples

			G.f. = 1 + 6480*q^4 + 157680*q^6 + ...
		

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, A362875, A362876, A362877, A362879, A362880.

Programs

Magma

prec := 20;
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];
S := SymmetricMatrix(ls);
L := LatticeWithGram(S);
T := ThetaSeries(L, 8);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..4]]);