OFFSET
0,4
COMMENTS
From Robert Coquereaux, Aug 05 2017: (Start)
Other avatars of {D_6}^{+} and its theta series:
The lattice L4 generated by cuts of the complete graph on a set of 4 vertices (rescaled by sqrt(2)).
The generalized laminated lattice Lambda_6[3] with minimal norm 3.
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.
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).
The theta function of {D_6}^{+} is b1 + 32 b4 + 60 b5.
(End)
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
M. Deza and V. Grishukhin, Delaunay Polytopes of Cut Lattices, Linear Algebra and Its Applications, 226- 228:667-685 (1995).
A. Ocneanu, The Classification of subgroups of quantum SU(N), 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.
W. Plesken and M. Pohst, Constructing integral lattices with prescribed minimum, Mathematics of Computation, Vol 45, No 171, pp. 209-221, and supplement S5-S16 (1985).
FORMULA
Expansion of (theta_2(q)^6 + theta_3(q)^6 + theta_4(q)^6)/2. - Seiichi Manyama, Oct 21 2018
EXAMPLE
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
MATHEMATICA
order = 50; S = (1/2) Series[
EllipticTheta[2, 0, q^2]^6 + EllipticTheta[3, 0, q^2]^6 +
EllipticTheta[4, 0, q^2]^6, {q, 0, order}];
CoefficientList[Simplify[Normal[S], Assumptions -> q > 0], q] (* Robert Coquereaux, Aug 05 2017 *)
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 *)
PROG
(Magma)
order:=50; // Example
H := DirichletGroup(16, CyclotomicField(EulerPhi(16)));
chars := Elements(H); eps := chars[2];
M := ModularForms([eps], 3);
Eltseq(PowerSeries(M![1, 0, 0, 32, 60, 0, 0], order)); // Robert Coquereaux, Aug 05 2017
(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 */
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved