A028997 Theta series of quadratic form with Gram matrix [ 4, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -1; 1, 0, -1, 4 ].

1, 0, 8, 8, 16, 8, 24, 0, 40, 16, 40, 16, 72, 24, 8, 32, 80, 16, 88, 24, 104, 8, 80, 32, 152, 48, 88, 48, 16, 48, 160, 48, 168, 64, 128, 8, 224, 48, 136, 64, 232, 48, 24, 48, 208, 104, 160, 80, 328, 0, 200, 112, 248, 64, 272, 96, 40, 112, 192, 88, 416, 72, 208, 16, 336, 112, 320

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1). - Michael Somos, Nov 22 2007

Examples

			G.f. = 1 + 8*x^2 + 8*x^3 + 16*x^4 + 8*x^5 + 24*x^6 + 40*x^8 + 16*x^9 + 40*x^10 + ...
G.f. = 1 + 8*q^4 + 8*q^6 + 16*q^8 + 8*q^10 + 24*q^12 + 40*q^16 + 16*q^18 + 40*q^20 + ...
		

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A030187, A134782.

Programs

Magma

A := Basis( ModularForms( Gamma0(14), 2), 67); A[1] + 8*A[3] + 8*A[4]; /* Michael Somos, Apr 19 2015 */

Mathematica

a[ n_] := With[{e1 = QPochhammer[ x] QPochhammer[ x^7], e2 = QPochhammer[ x^2] QPochhammer[ x^14]}, SeriesCoefficient[ e1^4 / e2^2 + 4 x e1 e2 + 8 x^2 e2^4 / e1^2, {x, 0, n}]]; (* Michael Somos, Apr 19 2015 *)

PARI

{a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^14 + A); A = eta(x + A) * eta(x^7 + A); polcoeff( A^4 / B^2 + 4 * x * A * B + 8 * x^2 * B^4 / A^2, n))}; /* Michael Somos, Nov 22 2007 */

PARI

{a(n) = my(G); if( n<0, 0, G = [ 4, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -1; 1, 0, -1, 4 ]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Nov 22 2007 */

Formula

G.f. is Fourier series of a weight 2 level 14 modular form. f(-1 / (14 t)) = 14 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 22 2007

Convolution of A030187 and A134782. - Michael Somos, Nov 22 2007