Dear David, The Sturm bound tells us how many coefficients we need to check before we know that two modular forms are the same (if the first B Fourier coefficients are the same then they're all the same). Maeda's conjecture tells us that we only need to check two coefficients in level 1 (otherwise the Hecke polynomial has a repeated eigenvalue, contradiction). I'm trying to get data for how many coefficients we need to check for newforms.
So say (x^2 + 1)^2 is the Hecke polynomial for T2. This gives a 2-dimensional i-eigenspace and a 2-dimensional (-i)-eigenspace. So there could be two newforms that look like q+i*q^2+... and two newforms that look like q-i*q^2+...; so one way now would be to take T3 acting on the i-eigenspace and T3 acting on the (-i)-eigenspace, and continue. Suppose (x-2)^2 is the Hecke polynomial for T3 acting on Kernel(T2^2 +1). This does not allow me to distinguish q+iq^2+2q^3+... and q-iq^2+2q^3+... I was referring to both approaches that you outlined. They both continue working with Kernel(T2^2 +1), say. Ideally, I'd like to consider T3 acting on Kernel(T+i) and Kernel(T-i) separately. This is more intuitive and would give me an exact answer, i.e. the least B such that if the first B Fourier coefficients are the same then they're all the same). I take the dimension 2 subspace of newforms of the type C*(q+iq^2+...), and use T3 to check the q^3 coefficient, and continue. Separately, I treat those of the form C*(q-iq^2+...) using T3. I hope that clarifies the problem. Thanks. Sam On Thursday, June 21, 2012 1:28:39 AM UTC+10, David Loeffler wrote: > > On 20 June 2012 15:23, Sam Chow <sam.cho...@gmail.com> wrote: > > 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. > > Dear Sam, > > Could you be more specific about exactly what you're trying to do > here? Are you referring to the first approach I outlined (working over > QQ) or the second approach (working over QQbar using explicit > subspaces of free modules and the hecke_matrix() method)? > > Regards, 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 URL: http://www.sagemath.org
