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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.

%I A004033 M5472 #41 Feb 16 2025 08:32:28
%S A004033 1,0,720,13440,97200,455040,1714320,4821120,12380400,29043840,
%T A004033 58980960,114076800,219310320,367338240,621878400,1037727360,
%U A004033 1583679600,2401816320,3747180240,5232470400,7551983520,10938261120,14715224640,19930775040,28073386800,35727920640
%N 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.
%C A004033 Also theta series of 16-dimensional lattice (SL(2,9) Y SL(2,9)).(C2 x C2). - _John Cannon_, Jan 10 2007
%C A004033 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A004033 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A004033 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%D A004033 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004033 G. C. Greubel, <a href="/A004033/b004033.txt">Table of n, a(n) for n = 0..1000</a>
%H A004033 Walter Feit, <a href="http://dx.doi.org/10.1016/0021-8693(78)90273-9">Some lattices over Q(sqrt(-3))</a>, J. Algebra 52 (1978), no. 1, 248-263.
%H A004033 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A004033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A004033 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.
%F A004033 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
%F A004033 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
%e A004033 G.f. = 1 + 720*x^2 + 13440*x^3 + 97200*x^4 + 455040*x^5 + 1714320*x^6 + 4821120*x^7 + ...
%e A004033 G.f. = 1 + 720*q^4 + 13440*q^6 + 97200*q^8 + 455040*q^10 + 1714320*q^12 + 4821120*q^14 + ...
%t A004033 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 *)
%o A004033 (Magma)
%o A004033 // Definition for lattice (SL(2,9) Y SL(2,9)).(C2 x C2), from _John Cannon_
%o A004033 LatticeWithBasis(16, \[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%o A004033 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
%o A004033 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], MatrixRing(IntegerRing(), 16) ! \[
%o A004033 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2,
%o A004033 1, -1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 1, 1, 2, 1, 0, 1,
%o A004033 1, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 4,
%o A004033 1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2,
%o A004033 0, 2, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1,
%o A004033 2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 1, 0, 1, -1, 1, 0, 1, 2, 0, 1, 4, 1,
%o A004033 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 4, 2, 2, 1, 0, 0, -1, 2,
%o A004033 1, 2, 1, 0, 2, 2, 2, 1, 2, 4, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2,
%o A004033 2, 2, 2, 4, 1, 0, -1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 1,
%o A004033 1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1,
%o A004033 1, 1, 1, 0, 1, -1, 1, 2, 4, 1, 1, 1, 2, 0, 2, 2, 1, 0, 1, -1, 1, 1, 1,
%o A004033 1, 1, 4 ])
%o A004033 (Magma)
%o A004033 // Definition for lattice A_2 tensor E_8, from _John Cannon_
%o A004033 A := Lattice("A", 2);
%o A004033 B := Lattice("E", 8);
%o A004033 L := TensorProduct(A, B);
%o A004033 T<q> := ThetaSeries(L, 16);
%o A004033 (Magma) A := Basis( ModularForms( Gamma0(3), 8), 26); A[1] + 720*A[3]; /* _Michael Somos_, Feb 01 2017 */
%o A004033 (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 */
%Y A004033 Cf. A004016, A007332, A037150.
%K A004033 nonn,easy
%O A004033 0,3
%A A004033 _N. J. A. Sloane_