A362880 Theta series of 20-dimensional lattice Kappa_20.

1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478, 69157229760, 151652810580, 311116423500, 607158951120, 1127694969072, 2020055770530, 3478103852940, 5829999042420, 9467119804680, 15046034533560

Offset: 0

Author

Andy Huchala, May 08 2023

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 15390*q^4 + 575160*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, A362878, A362879.

Programs

Magma

prec := 40;
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];
S := SymmetricMatrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
coeffs := [1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478];
Coefficients(&+[coeffs[i]*B[i] :i in [1..11]]);