Dear David, Sorry, I must have made a mistake when I was testing your second suggestion, and misunderstood the output. It does appear to do exactly what I wanted.
Thanks! Sam On Thursday, 21 June 2012 10:01:46 UTC+10, Sam Chow wrote: > > 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
