Theta series for modular lattices

[Sean A. Irvine]

Hi,

This month I would like to draw attention to another big block of sequences lacking in detail:

Provide formulas and/or programs for the sequences from A034599 to A034658. All of these sequences give the Theta series for various lattices and putative lattices described in a single paper by H.-G. Quebbemann (available via the Link in the corresponding sequences).

I track these questions here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[D. S. McNeil]

Disclaimer: this is right at the limit of "can kind of manage, but only with a tutorial open" for me. ;-) 

So I think I can match the values for those sequences programmatically in Sage, doing something like the below, but while the values match I'm not 100% certain I'm following the paper in doing so.  Hopefully some lattice expert can weigh in on whether the approach is sound.  Basically:
  
  Theta = theta series of the smallest l-modular root lattice, (D4 if l=2, A2 if l=3), with weight k0 (2 or 1).

  Delta_l = cusp form (eta(q) * eta(l*q))^(k1), where k1 = 24 / (1 + l) (8 or 6).

  Lattice dim is n, so total weight k = n/2.

  We can then form all the products Theta^lam * Delta_l^mu with k0 * lam + k1 * mu = k. 
  They should span the space of weight k forms at level l.

  For the extremal gap we set g=floor(k/k1), and solve for the coeffs where the q-expansion is 1 followed by g 0s.

So for a specific case, it looks like:

------------------------------------------------------------------------
Checking A034658: Theta series of (putative) extremal 3-modular even lattice in dimension 96.
------------------------------------------------------------------------
Extremal theta: level=3, dimension n=96, weight k=48
Using k0=1 (weight of Theta), k1=6 (weight of Delta_l)
Precision request: prec=50, head=400 (N=prec+head=450)
Built Theta=Theta_A2 and Delta_l:
  Theta head: 1 + 6*q + 6*q^3 + 6*q^4 + 12*q^7 + 6*q^9 + O(q^11)
  Delta head: 1*q + -6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 + -54*q^6 + -40*q^7 + 168*q^8 + 81*q^9 + -36*q^10 + O(q^11)
Monomials of total weight 48 (Theta^lam * Delta^mu):
  (lam=0, mu=8), (lam=6, mu=7), (lam=12, mu=6), (lam=18, mu=5), (lam=24, mu=4), (lam=30, mu=3), (lam=36, mu=2), (lam=42, mu=1), (lam=48, mu=0)
Extremal gap length g=floor(k/k1)=8; enforcing a[1..g]=0
Linear system matrix: shape = 8 x 9, rank = 8
Kernel dimension = 1 (expect 1)
Solved normalized combination (so a[0]=1):
  f = (-7921605600)*Delta^8 + (-75271680)*Theta^6*Delta^7 + (372726144)*Theta^12*Delta^6 + (-210325248)*Theta^18*Delta^5 + (28576800)*Theta^24*Delta^4 + (-1411200)*Theta^30*Delta^3 + (30240)*Theta^36*Delta^2 + (-288)*Theta^42*Delta + (1)*Theta^48
Gap check a[1..8] all zero? True
Comparing against OEIS values:
  ours: [1, 0, 0, 0, 0, 0, 0, 0, 0, 2370618432000, 272651888920320, 24922876620096000, 1479910662908496000, 63745506094548096000, 2075221360603119456000, 53130842241383142512640, 1103303415283892173260000, 19061424787521561471936000]
  ref : [1, 0, 0, 0, 0, 0, 0, 0, 0, 2370618432000, 272651888920320, 24922876620096000, 1479910662908496000, 63745506094548096000, 2075221360603119456000, 53130842241383142512640, 1103303415283892173260000, 19061424787521561471936000]
Status: match


Doug