%I #24 May 19 2024 08:06:42

%S 1,2,0,0,6,2,14,4,10,14,16,4,12,6,18,22,18,14,14,14,24,24,34,20,40,18,

%T 32,16,36,24,36,16,50,48,36,34,74,0,40,42,60,16,58,34,44,40,44,24,96,

%U 30,64,50,64,36,98,58,80,54,48,52,124,36,72,64,74,60,66,52,80,60,92,52

%N Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ].

%C This is the 4-dimensional Elkies_B lattice.

%H John Cannon, <a href="/A045866/b045866.txt">Table of n, a(n) for n = 0..5000</a>

%H N. D. Elkies, <a href="https://people.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.

%e 1 + 2*q^2 + 6*q^8 + 2*q^10 + 14*q^12 + 4*q^14 + 10*q^16 + 14*q^18 + 16*q^20 + ...

%o (PARI) B(x, y, z, w)=2*x^2+8*y^2+10*z^2+12*w^2+2*x*(y+w)+2*y*(z-3*w)+4*z*w;

%o thetaB(n, N, bx, by, bz, bw, ix, iy, iz, iw, nn)=n=2*n; bx=floor(sqrt(n)*(1+1/sqrt(6))); bz=floor(sqrt(n/7)); bw=floor(sqrt(n/6)); by=floor(sqrt(n/3)); N=vector(n/2+2); for(ix=-bx, bx, for(iy=-by, by, for(iz=-bz, bz, for(iw=-bw, bw, nn=B(ix, iy, iz, iw); if (nn<=n, N[1+nn/2]++); )))); N;

%o thetaB(80)

%o (PARI) {a(n)=if(n<1, n==0, qfrep([ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ], n, 1)[n]*2)} /* _Michael Somos_, Apr 02 2006 */

%Y Dual lattice to that in A045867.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000