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
Keywords
Examples
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 + ...
Links
- 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
Programs
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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 */
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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 *) -
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 */
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
Comments