A007038 - OEIS

A007038

Theta series of 20-dimensional lattice R_20 with det 1024 and minimal norm 4.

1, 0, 3960, 168960, 2094840, 16625664, 80652000, 343664640, 1073639160, 3299450880, 8000389584, 20081064960, 41278084320, 90307768320, 165298921920, 327406565376, 549720551160, 1009911767040, 1587054299160, 2748047385600, 4095742639824, 6764533923840 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`A034599 is almost certainly the same sequence. That will follow if R_20 is a 2-modular lattice. - N. J. A. Sloane, Sep 23 2019` `Theta series is a weight 10 modular form for Gamma_0(2). - Andy Huchala, May 13 2023`
	LINKS	`Andy Huchala, Table of n, a(n) for n = 0..20000` `C. Bachoc, Voisinage au sens de Kneser pour les réseaux quaternioniens, Comment. Math. Helvetici 70 (1995), 350-374.` `C. Bachoc and G. Nebe, Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order, Experiment. Math. 6 (1997), 151-162.` `G. Nebe and N. J. A. Sloane, Home page for this lattice`
	EXAMPLE	`G.f. = 1 + 3960q^4 + 168960q^6 + ...`
	PROG	`(Magma)` `prec := 10;` ls := [4, 0, 4, 0, 0, 4, 2, 2, 2, 4, 0, 0, -2, -1, 4, 0, 0, 0, 1, 0, 4, 2, 0, 0, 1, 0, 0, 4, 1, -1, -1, 0, 2, 2, 2, 4, 1, -1, -1, 0, 1, 1, 1, 1, 4, 1, 1, 1, 2, -1, 1, -1, -1, 0, 4, 1, -1, 1, 0, -1, 1, 1, 1, 0, 0, 4, 1, -1, 1, 1, 0, 2, 0, 1, 2, 2, 2, 4, -1, 1, 1, 0, -1, -1, -1, -2, -2, 0, 0, -1, 4, -1, -1, -1, -2, 1, -1, -1, 0, 0, -2, 0, -1, 0, 4, -1, 1, -1, 0, 1, 1, -1, 0, 0, 0, -2, -1, 0, 0, 4, -1, 1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -2, 2, 2, 2, 4, 1, -1, 1, 1, 0, 0, 0, 0, 1, 1, -1, 1, -2, 0, 0, -1, 4, 1, 1, 1, 1, 0, 0, 0, 0, -1, 1, 1, 1, 0, -2, 0, -1, 0, 4, -1, -1, 1, -1, 0, 0, 0, 0, 1, -1, 1, 1, 0, 0, -2, -1, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -2, 2, 2, 2, 4]; `S := SymmetricMatrix(ls);` `L := LatticeWithGram(S);` `M := ThetaSeriesModularFormSpace(L);` `B := Basis(M, prec);` `coeffs := [1, 0, 3960];` `Coefficients(&+[coeffs[i]*B[i] :i in [1..3]]); // Andy Huchala, May 13 2023`
	CROSSREFS	`Cf. A034599.` `Sequence in context: A319800 A373391 A034599 * A222783 A107540 A210787` `Adjacent sequences: A007035 A007036 A007037 * A007039 A007040 A007041`
	KEYWORD	`nonn`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`More terms from Andy Huchala, May 13 2023`
	STATUS	`approved`