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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 13 2014
EXTENSIONS
More terms from Andy Huchala, May 15 2023
STATUS
approved