Thanks for the reply, David. Your suggestions work well, in that I seem to end up with an exact result most of the time and a close result otherwise (compared to some weight 2 data by Stein).
I'll try to describe how the imprecision comes about. Say I get (x^2 + 1)^2. Ideally, I'd like to separate the i-eigenspace and the (-i)-eigenspace (this is for distinguishing Hecke eigenforms by looking at the first however many Fourier coefficients), and then continue with each of those separately. Combining those will give me an upper bound for how many primes I need to check, but not always an exact result, for instance q + i*q^2 + ... and q - i*q^2 + ... do not get distinguished by this particular Hecke operator (using this procedure). I can continue with the current procedure and get some results, however I'd still be very interested if you or anybody else knows a good way to separate eigenspaces within a Galois orbit. Sam On Thursday, June 14, 2012 12:25:04 AM UTC+10, Sam Chow wrote: > > I want to consider eigenspaces of S1 = > CuspForms(Gamma0(N),k).new_subspace(), but only for repeated > eigenvalues. So, given a Hecke eigenform T, and a repeated eigenvalue e, > I'd like to take the kernel of T-e acting on S1. This doesn't work because > elements of S1 must have rational coefficients, and e need not be rational, > so e might not act on S1. I'd like to do CuspForms(Gamma0(N),k,base_ring = > QQbar).new_subspace(), but there's a NotImplementedError. One solution is > to use S1 to get the repeated eigenvalues, and then get an "upper bound" > for the kernel I want by taking the kernel of T-e acting on S2. This is > okay for small S2, but if N and k have a lot of factors then the dimension > of S2 is high, and the matrices get very large. I've tried to shrink S2 > artificially by checking coefficients of the basis elements to see if > they're eigenforms, but I'm having difficulty putting certain basis > elements together to get a subspace. > > - What's going on with the NotImplementedError? > - Is there a solution, perhaps not using newforms but instead using > general eigenforms? > - Could I somehow take S1 and change the base ring, either directly or > manually (putting the basis vectors as a module over say QQbar)? > - If I use numerical eigenforms, I can't seem to get the Hecke > operators to work. Is there a way? > - I encountered similar problems when I used L instead of QQbar, where > the number field L is gotten by adjoining the eigenvalues. Has anybody > done > something like this before? > - Any other ideas? > > Thanks. > > > -- 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 URL: http://www.sagemath.org
