It's not so much that I haven't read the documentation, but rather that the definitions have been unclear. It's not Sage, it's me--things have been unclear all my life. To paraphrase from 1969's "Both sides now", I really don't know "numerical_eigenforms" at all. What Sage calls a cusp form on Gamma1 (N) with nebentypus is what I have called, since childhood, a cusp form on Gamma0(N) with character multiplier. I blame my parents. The command n=numerical_eigenforms(Gamma0(N),k) evidently deals with the case of trivial nebentypus, which of course explains why n.ap(2) returns an empty list when k is odd. On the other hand, if I'm thinking correctly, the command n=numerical_eigenforms(Gamma1(N),k) should handle *all* the nebentypuses (sorry if that's the incorrect plural), i.e., all Dirichlet characters mod N. To see if I have the slightest idea what is going on, I wrote this one line Sage program:
sage: numerical_eigenforms(Gamma1(11),2).systems_of_abs(14) [ [1.3281310261, 3.44297864737, 4.23200071553, 7.37063348168, 1.0, 12.2051447409], [1.3281310261, 3.44297864737, 4.23200071553, 7.37063348168, 1.0, 12.2051447409], [1.3281310261, 3.44297864737, 4.23200071553, 7.37063348168, 11.0, 12.2051447409], [1.3281310261, 3.44297864737, 4.23200071553, 7.37063348168, 11.0, 12.2051447409], [2.0, 1.0, 1.0, 2.0, 1.0, 4.0], [2.49721204096, 2.2684571924, 5.3935303785, 6.21882320691, 11.0, 13.3429547647], [2.49721204096, 2.2684571924, 5.3935303785, 6.21882320691, 11.0, 13.3429547647], [2.49721204096, 2.2684571924, 5.3935303785, 6.21882320691, 1.0, 13.3429547647], [2.49721204096, 2.2684571924, 5.3935303785, 6.21882320691, 1.0, 13.3429547647], [3.0, 4.0, 6.0, 8.0, 1.0, 14.0] ] Fine, the fifth item in this list corresponds to something even I can understand--it's the weight 2 newform on Gamma0(11) which is simply a product of four eta functions, and it can be found on the opening page of Stein's Modular Forms Explorer. The last item in the list I kinda understand, but I'm not really sure how it corresponds to a standard "eigenform"; the eigenvalues a(p) (except for p=11) are the coefficients of the (nonholomorphic) Eisenstein series of weight 2 (sorta akin to a weight 2 oldform on the full modular group). The other eight items in the list are--well, I dunno. One might guess that items 1-4 correspond to four Galois conjugate eigenforms with a quintic nebentypus mod 11, and similarly for items 6-9. But there are no such weight 2 "eigenforms" with quintic nebentypus in the space of cusp forms, are there? One certainly won't find them in the Modular Forms Explorer. So what are these animals? I leave with the following challenge: Use Sage to determine (algebraically, not just numerically) the first ten coefficients c(p) of the "eigenform" corresponding to item 1 in the list. rje p.s. Just to make sure there is at least one thing useful in this post, I point out a misprint after the word "polynomial" at http://www.sagemath.org/doc/bordeaux_2008/level_one_forms.html ********************************************************************************************************************************** On Jul 19, 12:18 pm, William Stein <wst...@gmail.com> wrote: > 2009/7/15rje<rev...@ucsd.edu>: > > > > > What is going on here? Does this only work for even weights?rje > > > sage: n=numerical_eigenforms(15,3);n.ap(2) > > [] ************************************************************************************** > Type > > sage: numerical_eigenforms? > > and read it. In particular, the first input is the group and if a > number N is given that it just defaults to Gamma0(N). So the above is > correctly giving you the eigenvalues for Gamma0(15) and odd weight. > > sage: n=numerical_eigenforms(Gamma1(15),3);n.ap(2) > [-5.0 - 1.99021290795e-14*I, -3.0 + 2.40206990184e-13*I, 5.0 + > 3.72521051072e-12*I, 1.00000000001 - 4.0*I, 0.999999999998 + 4.0*I, > -3.0 - 3.86606026001e-12*I, 0.224744871392 - 0.224744871392*I, > 0.224744871392 + 0.224744871392*I, -2.22474487139 - 2.22474487139*I, > 1.0 - 4.0*I, -3.48610029732e-14 - 2.2360679775*I, -1.0 + > 1.43583146453e-14*I, -4.27435864481e-14 + 2.2360679775*I, 1.0 + 4.0*I, > -2.22474487139 + 2.22474487139*I, 1.0 - 1.21528935033e-14*I] > > William --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
