A029713 Theta series of 6-dimensional 8-modular lattice of minimal norm 4.

1, 0, 30, 56, 66, 144, 188, 288, 378, 448, 528, 504, 884, 1008, 1056, 1440, 1290, 1344, 1834, 1848, 2064, 2880, 2652, 3168, 3332, 2688, 3696, 3696, 4128, 5040, 5280, 5760, 5610, 5824, 6012, 5376, 7798, 8208, 7164, 10080, 8208, 8064, 10560, 8568, 10068

Offset

0,3

Comments

Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2. - Michael Somos, Nov 24 2007

Links

Table of n, a(n) for n=0..44.

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

Formula

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (512)^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 24 2007

A136028(n) = a(n) + 6 * A030207(n). - Michael Somos, Apr 19 2015

Example

G.f. = 1 + 30*x^2 + 56*x^3 + 66*x^4 + 144*x^5 + 188*x^6 + 288*x^7 + ...
G.f. = 1 + 30*q^4 + 56*q^6 + 66*q^8 + 144*q^10 + 188*q^12 + 288*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^8], e2 = QPochhammer[ x^2] QPochhammer[ x^4]}, e2^9 / e1^6 - 6 x e1^2 e2], {x, 0, n}]; (* Michael Somos, Apr 19 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A) * eta(x^4 + A) )^9 / ( eta(x + A) * eta(x^8 + A) )^6 - 6 * x * ( eta(x + A) * eta(x^8 + A) )^2 * eta(x^2 + A) * eta(x^4 + A), n))}; /* Michael Somos, Nov 24 2007 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [4, 1, -1, -1, 1, -1; 1, 4, 0, 1, 2, 1; -1, 0, 4, -1, 2, -1; -1, 1, -1, 4, -1, 0; 1, 2, 2, -1, 4, -1; -1, 1, -1, 0, -1, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Nov 24 2007 */
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 45); A[1] + 30*A[3] + 56*A[4] + 66*A[5] + 144*A[6] + 188*A[7]; /* Michael Somos, Apr 19 2015 */

Crossrefs

Cf. A030207, A136028.

Sequence in context: A006315 A027578 A189712 * A154599 A048451 A154796

Adjacent sequences: A029710 A029711 A029712 * A029714 A029715 A029716

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved