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Theta series of 20-dimensional lattice R_20 with det 1024 and minimal norm 4.
2

%I #29 May 13 2023 06:21:40

%S 1,0,3960,168960,2094840,16625664,80652000,343664640,1073639160,

%T 3299450880,8000389584,20081064960,41278084320,90307768320,

%U 165298921920,327406565376,549720551160,1009911767040,1587054299160,2748047385600,4095742639824,6764533923840

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

%C 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

%C Theta series is a weight 10 modular form for Gamma_0(2). - _Andy Huchala_, May 13 2023

%H Andy Huchala, <a href="/A007038/b007038.txt">Table of n, a(n) for n = 0..20000</a>

%H C. Bachoc, <a href="https://doi.org/10.1007/BF02566012">Voisinage au sens de Kneser pour les réseaux quaternioniens</a>, Comment. Math. Helvetici 70 (1995), 350-374.

%H C. Bachoc and G. Nebe, <a href="http://www.emis.de/journals/EM/expmath/volumes/6/6.html">Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order</a>, Experiment. Math. 6 (1997), 151-162.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/R20.html">Home page for this lattice</a>

%e G.f. = 1 + 3960*q^4 + 168960*q^6 + ...

%o (Magma)

%o prec := 10;

%o 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];

%o S := SymmetricMatrix(ls);

%o L := LatticeWithGram(S);

%o M := ThetaSeriesModularFormSpace(L);

%o B := Basis(M,prec);

%o coeffs := [1,0,3960];

%o Coefficients(&+[coeffs[i]*B[i] :i in [1..3]]); // _Andy Huchala_, May 13 2023

%Y Cf. A034599.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Andy Huchala_, May 13 2023