A004033 Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice.

1, 0, 720, 13440, 97200, 455040, 1714320, 4821120, 12380400, 29043840, 58980960, 114076800, 219310320, 367338240, 621878400, 1037727360, 1583679600, 2401816320, 3747180240, 5232470400, 7551983520, 10938261120, 14715224640, 19930775040, 28073386800, 35727920640

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also theta series of 16-dimensional lattice (SL(2,9) Y SL(2,9)).(C2 x C2). - John Cannon, Jan 10 2007

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 720*x^2 + 13440*x^3 + 97200*x^4 + 455040*x^5 + 1714320*x^6 + 4821120*x^7 + ...
G.f. = 1 + 720*q^4 + 13440*q^6 + 97200*q^8 + 455040*q^10 + 1714320*q^12 + 4821120*q^14 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Walter Feit, Some lattices over Q(sqrt(-3)), J. Algebra 52 (1978), no. 1, 248-263.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A004016, A007332, A037150.

Programs

Magma

// Definition for lattice (SL(2,9) Y SL(2,9)).(C2 x C2), from John Cannon
LatticeWithBasis(16, \[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], MatrixRing(IntegerRing(), 16) ! \[
4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2,
1, -1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 1, 1, 2, 1, 0, 1,
1, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 4,
1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2,
0, 2, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1,
2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 1, 0, 1, -1, 1, 0, 1, 2, 0, 1, 4, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 4, 2, 2, 1, 0, 0, -1, 2,
1, 2, 1, 0, 2, 2, 2, 1, 2, 4, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2,
2, 2, 2, 4, 1, 0, -1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 1,
1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1,
1, 1, 1, 0, 1, -1, 1, 2, 4, 1, 1, 1, 2, 0, 2, 2, 1, 0, 1, -1, 1, 1, 1,
1, 1, 4 ])

Magma

// Definition for lattice A_2 tensor E_8, from John Cannon
A := Lattice("A", 2);
B := Lattice("E", 8);
L := TensorProduct(A, B);
T := ThetaSeries(L, 16);

Magma

A := Basis( ModularForms( Gamma0(3), 8), 26); A[1] + 720*A[3]; /* Michael Somos, Feb 01 2017 */

Mathematica

a[ n_] := SeriesCoefficient[ With[ {a1 = (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3]}, a1^2 (a1^6 - 48 x QPochhammer[ x]^6 QPochhammer[ x^3]^6)], {x, 0, n}]; (* Michael Somos, Feb 01 2017 *)

PARI

{a(n) = if( n<0, 0, my(A, a1); A = x * O(x^n); a1 = (eta(x + A)^3 + 9*x * eta(x^9 + A)^3) / eta(x^3 + A); polcoeff( a1^2 * (a1^6 - 48*x * eta(x + A)^6 * eta(x^3 + A)^6), n))}; /* Michael Somos, Feb 01 2017 */

Formula

Theta series is x^8-48*x^2*y, x = phi_0(z) (see A004016), y = Delta_12(z) (see A007332) in the notation of SPLAG, Chap. 4. See A037150 for Maple code.

Expansion of a(x)^2 * (a(x)^6 - 48*x * f(-x)^6 * f(-x^3)^6) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017