%I #41 Apr 24 2023 11:20:41

%S 1,0,0,32,60,0,0,192,252,0,0,480,544,0,0,832,1020,0,0,1440,1560,0,0,

%T 2112,2080,0,0,2624,3264,0,0,3840,4092,0,0,4992,4380,0,0,5440,6552,0,

%U 0,7392,8160,0,0,8832,8224

%N Theta series of {D_6}^{+} lattice.

%C From _Robert Coquereaux_, Aug 05 2017: (Start)

%C Other avatars of {D_6}^{+} and its theta series:

%C The lattice L4 generated by cuts of the complete graph on a set of 4 vertices (rescaled by sqrt(2)).

%C The generalized laminated lattice Lambda_6[3] with minimal norm 3.

%C The first member (k=1) of the family of lattices of SU(3) hyper-roots associated with the fusion category A_k(SU(3)); simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k. This lattice has to be rescaled: q --> q^2 since its minimal norm is 6 whereas the minimal norm of {D_6}^{+} is 3.

%C The space of modular forms on Gamma_1(16) of weight 3, twisted by a Dirichlet character defined as the Kronecker character -4, has dimension 7 and basis b1,...b7, where bn has leading term q^(n-1).

%C The theta function of {D_6}^{+} is b1 + 32 b4 + 60 b5.

%C (End)

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

%H Seiichi Manyama, <a href="/A008434/b008434.txt">Table of n, a(n) for n = 0..10000</a>

%H Robert Coquereaux, <a href="https://arxiv.org/abs/1708.00560">Theta functions for lattices of SU(3) hyper-roots</a>, arXiv:1708.00560 [math.QA], 2017.

%H M. Deza and V. Grishukhin, <a href="https://doi.org/10.1016/0024-3795(95)00240-R">Delaunay Polytopes of Cut Lattices</a>, Linear Algebra and Its Applications, 226- 228:667-685 (1995).

%H A. Ocneanu, <a href="https://cel.archives-ouvertes.fr/cel-00374414">The Classification of subgroups of quantum SU(N)</a>, in "Quantum symmetries in theoretical physics and mathematics", Bariloche 2000, Eds. Coquereaux R., Garcia A. and Trinchero R., AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.

%H W. Plesken and M. Pohst, <a href="https://doi.org/10.1090/S0025-5718-1993-1176715-1">Constructing integral lattices with prescribed minimum</a>, Mathematics of Computation, Vol 45, No 171, pp. 209-221, and supplement S5-S16 (1985).

%F Expansion of (theta_2(q)^6 + theta_3(q)^6 + theta_4(q)^6)/2. - _Seiichi Manyama_, Oct 21 2018

%e G.f. = 1 + 32*q^3 + 60*q^4 + 192*q^7 + 252*q^8 + 480*q^11 + 544*q^12 + ... - _Michael Somos_, Sep 09 2018

%t order = 50; S = (1/2) Series[

%t EllipticTheta[2, 0, q^2]^6 + EllipticTheta[3, 0, q^2]^6 +

%t EllipticTheta[4, 0, q^2]^6, {q, 0, order}];

%t CoefficientList[Simplify[Normal[S], Assumptions -> q > 0], q] (* _Robert Coquereaux_, Aug 05 2017 *)

%t a[ n_] := With [{e1 = QPochhammer[ q^2]^12, e2 = QPochhammer[ q^4]^6, e3 = QPochhammer[ q^8]^12}, SeriesCoefficient[ (e2^6 + e1 e3 (e1 + 64 q^3 e3)) / (2 e1 e2 e3), {q, 0, n}]]; (* _Michael Somos_, Sep 09 2018 *)

%o (Magma)

%o order:=50; // Example

%o H := DirichletGroup(16,CyclotomicField(EulerPhi(16)));

%o chars := Elements(H); eps := chars[2];

%o M := ModularForms([eps],3);

%o Eltseq(PowerSeries(M![1,0,0,32,60,0,0],order)); // _Robert Coquereaux_, Aug 05 2017

%o (Magma) A := Basis( ModularForms( Gamma1(16), 3), 50); A[1] + 32*A[4] + 60*A[5] + 192*A[8] + 252*A[9] + 480*A[12] + 544*A[13] + 832*A[16] + 1020*A[17] + 1440*A[20] + 1560*A[21]; /* _Michael Somos_, Sep 09 2018 */

%o (PARI) {a(n) = my(A, e1, e2, e3); if( n<0, 0, A = x * O(x^n); e1 = eta(x^2)^12; e2 = eta(x^4 + A)^6; e3 = eta(x^8 + A)^12; polcoeff( (e2^6 + e1*e3*(e1 + 64 * x^3 * e3)) / (2 * e1 * e2 * e3), n))}; /* _Michael Somos_, Sep 09 2018 */

%Y Cf. A290654, A290655, A287329, A287944, A288488, A288489, A288776, A288779, A288909.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_