A056945 Jacobi form of weight 12 and index 1 associated to a (nonexistent) lattice vector of norm 2 for the Leech lattice.

1, 0, 0, -4, 6, 0, 0, 32736, 131076, 0, 0, 3669012, 9172952, 0, 0, 95691552, 188239518, 0, 0, 1142929524, 1959705000, 0, 0, 8506686816, 13293227112, 0, 0, 45763087664, 67073100864, 0, 0, 195387947712, 272567759508, 0, 0, 698077783656, 938807478318, 0, 0, 2176654050912

Offset

0,4

Comments

Let J(h)=E_8*E_{4,1}+(2h-60)*phi_{12,1} be the Jacobi form of weight 12 and index 1 associated with a norm 2 vector of a Niemeier lattice of Coxeter number h. Let J(h)=sum_{n,r} c(4n-r^2) q^n*z^r. So a(n)=c(4m-r^2) for h=0.

Let N(h,n) be the number of vectors of norm 2n for the lattice, then we have N(h,n)=c(4n)+2*sum_{1<=r<=sqrt(4n)}c(4n-r^2) if h is the Coxeter number of a Niemeier lattice. Note that N(0,n)=a(4n)-2*sum a(4n-r^2)=A008408(n), for the Leech lattice! Note also a(3)<0 and a(n) is nonnegative for n<=1000, except 3.

References

Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Links

Table of n, a(n) for n=0..39.

Formula

E_8*E_{4, 1}-60*phi_{12, 1}. The E's are Eisenstein-Jacobi series and phi_{12, 1} is the unique normalized Jacobi cusp form of weight 12 and index 1.

a(n) = A055747(n) - 60*A003785(n). - Sean A. Irvine, May 18 2022

Crossrefs

Cf. A003785, A008408, A056946, A055747.

Sequence in context: A079207 A259825 A276449 * A324472 A070683 A195785

Adjacent sequences: A056942 A056943 A056944 * A056946 A056947 A056948

Keyword

sign

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 16 2000

Status

approved