A125509 Theta series of 4-dimensional lattice QQF.4.f.

1, 0, 0, 12, 12, 0, 12, 0, 24, 12, 24, 24, 24, 12, 24, 24, 24, 24, 36, 24, 48, 48, 48, 0, 60, 48, 36, 36, 72, 36, 96, 12, 60, 24, 48, 48, 108, 48, 72, 84, 96, 60, 120, 48, 72, 72, 0, 60, 132, 48, 96, 96, 120, 48, 132, 96, 144, 72, 108, 48, 168, 96, 108, 96, 144, 72, 144, 72, 144, 12

Offset

0,4

Comments

The quadratic form associated with the lattice can be expressed as Q(x, y, z, w) = (2*x+z+w)^2 + (x+2*y)^2 + (x+y)^2 + (y+2*z)^2 + (z+2*w)^2 + w^2 which comes from the basis (2,1,1,0,0,0), (0,2,1,1,0,0), (1,0,0,2,1,0), (1,0,0,0,2,1). - Michael Somos, Mar 30 2015

Links

John Cannon, Table of n, a(n) for n = 0..5000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Example

G.f. = 1 + 12*x^3 + 12*x^4 + 12*x^6 + 24*x^8 + 12*x^9 + 24*x^10 + 24*x^11 + ...
G.f. = 1 + 12*q^6 + 12*q^8 + 12*q^12 + 24*q^16 + 12*q^18 + 24*q^20 + 24*q^22 + 24*q^24 + ...

Mathematica

a[ n_] := With[{B = QPochhammer[ x^2] QPochhammer[ x^46]}, With[{A = QPochhammer[ x] QPochhammer[ x^23] / B}, SeriesCoefficient[ (4 x^3 + 4 A x^2 + A^3) (4 x^3 + 4 A x^2 + 4 A^2 x + A^3) (B / A)^2, {x, 0, n}]]]; (* Michael Somos, Mar 23 2015 *)

Prog

(PARI)  {a(n) = my(G); if( n<0, 0, G = [ 6, 3, 2, 2; 3, 6, 2, 0; 2, 2, 6, 3; 2, 0, 3, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* Michael Somos, Mar 23 2015 */
(PARI) {a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^46 + A); A = eta(x + A) * eta(x^23 + A) / B; polcoeff( (4*x^3 + 4*A*x^2 + A^3) * (4*x^3 + 4*A*x^2 + 4*A^2*x + A^3) * (B / A)^2, n))}; /* Michael Somos, Mar 23 2015 */
(Magma) Basis( ModularForms( Gamma0(23), 2), 70)[1]; /* Michael Somos, Mar 23 2015 */
(Sage) A = ModularForms( Gamma0(23), 2, prec=70).basis(); A[2] - 12/11*(A[0] + 3*A[1]); # Michael Somos, Mar 23 2015

Crossrefs

Sequence in context: A195748 A038337 A155825 * A281251 A247511 A097824

Adjacent sequences: A125506 A125507 A125508 * A125510 A125511 A125512

Keyword

nonn

Author

N. J. A. Sloane, Jan 31 2007

Status

approved