> > Well, B[3] is f(q^4) where f is the form that Steve Finch is after.
> > His form is some linear combination of B[0], B[1] and B[3]. But we
> > don't have any nice way of identifying the level 4 things as a
> > subspace of the level 16 things; it's more or less just luck that the
> > fourth echelon basis vector happens to be f(q^4).
>
> Yep. I found that amusing :-) It would be great if the half
> integral forms code in Sage were to greatly improve. What is there
> now is pretty limited.
On reflection, there is an algorithm there: if N is not divisible by
16, and M = lcm(N, 16), then the space of half-integral weight modular
forms of level N must be exactly the subspace of the space of forms of
level M consisting of those f for which f(M/N q) is also in the space.
Here's the level 28 example:
B = half_integral_weight_modform_basis(DirichletGroup(4*28)(1), 3, 50)
q = B[0].parent().gen()
V = QQ^50
W = V.subspace([V(f(q).padded_list(50)) for f in B])
W4 = V.subspace([V(f(q**4).padded_list(50)) for f in B])
t = [W.coordinate_vector(x) for x in W4.intersection(W).basis()]
for i in xrange(len(t)):
F=sum([t[i][j] * B[j] for j in xrange(len(t[i]))])
G =sum([q**j*F[4*j] for j in xrange(F.prec()//4)])
print G
This outputs q - q^2 - q^4 + q^7 + q^8 - q^9. Similarly for the weight
5/2, level 28 and trivial character case we get the following 4 forms:
q + 14*q^2 + 100*q^3 - 744*q^4 - 100*q^5 + 599*q^7 + 2174*q^8 -
1431*q^9 - 200*q^10 + 1482*q^11
q^2 + 14*q^3 - 92*q^4 - 14*q^5 + 75*q^7 + 272*q^8 - 180*q^9 - 28*q^10
+ 184*q^11
q^3 - 2*q^4 - q^5 + q^7 + 6*q^8 - 4*q^9 - 2*q^10 + 4*q^11
q^4 - q^7 - 3*q^8 + 2*q^9 - 2*q^11
Hope this is of some use to you.
David
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-support@googlegroups.com
To unsubscribe from this group, send email to
sage-support-unsubscr...@googlegroups.com
For more options, visit this group at
http://groups.google.com/group/sage-support
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
