A037150 Fourier coefficients of Eisenstein series of degree 2 and weight 6 when evaluated at Gram(A_2)*z.

1, 0, -1512, 44352, 449064, 6519744, 47263608, 257027904, 1115041032, 4093040448, 13000566096, 37057027392, 96945887304, 232758852480, 526296318912, 1128198915648, 2286101175624, 4451375005056, 8386154766360, 15131349955008, 26614555499952

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third Ed., 1998.
• Helmut Klingen, Introductory Lectures on Siegel Modular Forms, Cambrifge, 1990, p. 123.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..100
• N. J. A. Sloane, Notes on Two-dimensional Theta Series of Lattices (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".
• Index entries for sequences related to Eisenstein series

Crossrefs

Cf. A004016, A007332.

Programs

Maple

# Maple code from N. J. A. Sloane, Dec 12 2020. Will also be useful for related sequences.
# get standard theta series in Maple. First set max degree, maxd.
maxd:=201:
# get th2, th3, th4 = Jacobi theta constants out to degree maxd (Ref. Conway-Sloane, p. 102)
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a,q,maxd);
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a,q,maxd);
th4:=series(subs(q=-q,th3),q,maxd);
# get Dedekind eta function
a:=q^(1/24) : for m from 1 to maxd do a:=a*(1-q^m); od:
eta:=a;
# get phi0 and phi1 (Ref. Conway-Sloane, p. 103)
phi0:=series( subs(q=q^2,th2)*subs(q=q^6,th2)+subs(q=q^2,th3)*subs(q=q^6,th3), q, maxd );
phi1:=series( subs(q=q^2,th2)*subs(q=q^6,th3)+subs(q=q^2,th3)*subs(q=q^6,th2), q, maxd );
# get delta12 (Ref. Conway-Sloane, p. 204, where it is called Delta_6)
delta12:=series((subs(q=q^3,eta)*eta)^6,q,maxd);
delta12:=series(subs(q=q^2,delta12),q,maxd);
# To get the present sequence: (Ref. Sloane notebook pages)
x:=phi0; y:=delta12;
w1:= x^12-72*x^6*y-1728*y^2; w1s:=series(w1,q,maxd); w2:=subs(q=sqrt(t),w1s); w3:=series(w2,t,101);
w4:=seriestolist(w3); # A037150

Formula

x^12-72*x^6*y-1728*y^2, x = phi_0(z), y = Delta_12(z). Here phi_0(z) is the theta series of the A_2 lattice - see A004016, and Delta_12 is the 12-dimensional cusp form given in A007332.

Extensions

Entry revised by N. J. A. Sloane, Dec 12 2020