A239917 Theta series of 16-dimensional lattice OBW16, an overlattice of the Barnes-Wall lattice BW16.

1, 0, 0, 512, 4320, 18432, 61440, 193536, 522720, 1126400, 2211840, 4584960, 8960640, 14764032, 23224320, 40221696, 67154400, 96546816, 135168000, 210332160, 319809600, 423976960, 550195200, 801119232, 1147643520, 1436147712, 1771683840, 2462397440, 3371915520

Offset

0,4

Comments

The 512 vectors of norm 3 form a spherical 5-design (see Neumaier, 1981). The corresponding configuration of 256 lines in 16-space was studied by Shult and Yanushka, 1980.

This theta series is an element of the space of modular forms on Gamma_0(4) of weight 8 and dimension 5. - Andy Huchala, May 15 2023

Links

Andy Huchala, Table of n, a(n) for n = 0..10000

G. Nebe and N. J. A. Sloane, Home page for this lattice

A. Neumaier, Combinatorial configurations in terms of distances, Report 81-09-Wiskunde, Tech. Univ. Eindhoven, 1981.

A. Neumaier, Lattices of simplex type, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 145--160. MR0699768 (85f:05040). See Example 3.

Ernest Shult, Arthur Yanushka, Arthur, Near n-gons and line systems, Geom. Dedicata 9 (1980), no. 1, 1--72. MR0566437 (82b:51018).

Example

The theta series is 1 + 512*q^3 + 4320*q^4 + 18432*q^5 + 61440*q^6 + 193536*q^7 + 522720*q^8 + 1126400*q^9 + 2211840*q^10 + 4584960*q^11 + 8960640*q^12 + 14764032*q^13 + 23224320*q^14 + 40221696*q^15 + 67154400*q^16 + O(q^17).

Prog

(Magma)
L:=LatticeWithGram(16, [3,
-1, 3,
-1, -1, 3,
1, -1, 1, 3,
0, 1, 0, -1, 3,
-1, 0, 0, -1, -1, 3,
1, 0, 0, 1, -1, -1, 3,
-1, 0, 1, 0, 1, -1, -1, 3,
1, 0, -1, 0, -1, 1, 1, -1, 3,
-1, 0, 1, 0, -1, 0, 0, 1, -1, 3,
0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 3,
1, -1, 1, 1, -1, 0, 1, -1, 0, 0, 0, 3,
-1, 0, 1, 1, 0, -1, 0, 1, -1, 1, 0, 0, 3,
0, 0, 1, 0, 1, 0, -1, 1, -1, 0, 0, 0, -1, 3,
1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1, 1, 3,
1, -1, -1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 3]);
T<q>:=ThetaSeries(L, 16);
T;

Crossrefs

Sequence in context: A186798 A017067 A186845 * A114287 A061209 A017259

Adjacent sequences: A239914 A239915 A239916 * A239918 A239919 A239920

Keyword

nonn

Author

N. J. A. Sloane, Apr 13 2014

Extensions

More terms from Andy Huchala, May 15 2023

Status

approved