was@form:~/168/notes/2005-09-26$ sage --------------------------------------------------------- SAGE Version 0.7.4, Export Date: 2005-09-27-0000 Distributed under the terms of the GNU General Public L IPython shell -- for help type ?, ??, % --------------------------------------------------------- sage: M = ModularSymbols(11) sage: M _2 = Full Modular Symbols space for Gamma_0(11) of weighign 0 and dimension 3 over Rational Field sage: M.basis() _3 = ((1,0), (1,8), (1,9)) sage: M = ModularSymbols(11,sign=1) sage: M.basis() _5 = ((1,0), (1,9)) sage: M([1,0]) --------------------------------------------------------- Traceback (most recent call last): File "", line 1, in ? File "/home/was/sage/local/lib/python2.4/site-packages/ar/modsym/ambient.py", line 214, in __call__ raise TypeError, "No coercion of %s into %s defined." TypeError: No coercion of [1, 0] into Full Modular Symbolr Gamma_0(11) of weight 2 with sign 1 and dimension 2 ove Field defined. sage: M.basis() _7 = ((1,0), (1,9)) sage: M.hecke_matrix(2) _8 = [ 3 -1] [ 0 -2] sage: M = ModularSymbols(37,sign=1) sage: M.basis() _10 = ((1,0), (1,23), (1,34)) sage: T7 = M.hecke_matrix(7) sage: T7 _12 = [ 8 0 -6] [ 0 -1 0] [ 0 0 -1] sage: charpoly(T7) _13 = x^3 - 6*x^2 - 15*x - 8 sage: factor(charpoly(T7)) _14 = (x - 8) * (x + 1)^2 sage: M.hecke_matrix(11) _15 = [12 0 -6] [ 0 -5 0] [ 0 0 3] sage: T13 = M.hecke_matrix(13) sage: T7 * T13 _17 = [112 0 -72] [ 0 2 0] [ 0 0 4] sage: T13 * T7 _18 = [112 0 -72] [ 0 2 0] [ 0 0 4] sage: factor(charpoly(T13)) _19 = (x - 14) * (x + 2) * (x + 4) sage: kernel(T13+2) _20 = Vector space of degree 3 and dimension 1 over Rational Fi Basis matrix: [0 1 0] sage: kernel(T13-14) _21 = Vector space of degree 3 and dimension 1 over Rational Fi Basis matrix: [ 1 0 -2/3] sage: M.basis() _22 = ((1,0), (1,23), (1,34)) sage: R = Integers(37); R(1/23) _23 = 29 sage: R(1/34) _24 = 12 sage: kernel(T13+4) _25 = Vector space of degree 3 and dimension 1 over Rational Fi Basis matrix: [0 0 1] sage: factor(charpoly(T13)) _26 = (x - 14) * (x + 2) * (x + 4) sage: v = kernel(T13+4).basis()[0] sage: v _28 = (0, 0, 1) sage: T17 = M.hecke_matrix(17) sage: v*T17 _30 = (0, 0, 6) sage: M = ModularSymbols(2,sign=1) sage: T13 = M.hecke_matrix(13) sage: T13 _33 = [14] sage: M = ModularSymbols(389,sign=1) sage: T2 = M.heck M.hecke_algebra M.hecke_module_of_level M.hecke_bound M.hecke_operator M.hecke_matrix M.hecke_polynomial sage: T2 = M.hecke_matrix(2) sage: T2 _36 = [ 3 0 -1 0 0 -1 1 0 0 0 -1 1 -1 0 1 1 0 1 -1 1 -1 1 0 0 0 0 1 -1 -1] [ 0 0 0 1/2 0 1/2 0 -3/2 1/2 0 0 0 0 1 -1/2 0 0 0 0 -1/2 1/2 -1 0 1 1/2 1/2 0 0 0] [ 0 0 0 0 0 -1 1 -1 1 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 -0 -1 -1 1 0 0 0] [ 0 0 0 -1 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 -1 1 0 0 0 0] [ 0 2 -1 -1 2 0 0 0 0 1 -2 -1 0 2 1 2 0 1 -2 1 -1 1 0 -1 0 1 0 -1 0] [ 0 1 1/2 -1/2 1 0 -1/2 1/2 -1/2 0 -3/2 1/2 -1/2 1 1 3/2 0 1/2 -3/2 1 0 1/2 -0 -1 -1/2 1/2 1/2 -1/2 -1/2] [ 0 2 0 -2 3 0 1 -1 1 2 -1 -1 1 2 0 1 1 -1 -1 0 0 -1 0 0 1 1 -1 1 -1] [ 0 2 1 -1 2 0 1 -1 1 2 -1 -1 1 2 -1 1 1 -1 -1 0 0 -2 0 1 2 1 -1 1 -1] [ 0 -2 -1 1 -2 0 -1 1 -1 -2 1 1 -1 -2 0 -1 0 1 2 0 -1 2 -1 0 -1 -1 2 -2 1] [ 0 -1 -1 1/2 -1 -1/2 -1 1/2 -3/2 -1 0 1 -1 -1 1/2 -1 1 0 1 1/2 -1/2 0 0 0 -1/2 1/2 0 0 0] [ 0 -1 -1 1 -1 0 0 0 0 -1 1 1 0 -2 0 -1 0 0 1 0 0 0 0 1 -1 0 0 0 0] [ 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -0 0 -1 1 -1 0 0] [ 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 -1 0 0] [ 0 -1 -1/2 1/2 -1 0 -1/2 1/2 -1/2 -1 1/2 1/2 -1/2 -1 0 -1/2 0 -1/2 1/2 0 1 -1/2 -0 0 -1/2 -1/2 -1/2 1/2 1/2] [ 0 0 0 -1 0 0 0 1 -1 0 0 1 0 0 1 0 0 0 0 1 0 -1 0 1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0] [ 0 0 0 1/2 0 1/2 0 -1/2 1/2 0 0 0 0 0 -1/2 0 1 -1 0 -1/2 1/2 0 -0 -1 1/2 -1/2 0 0 0] [ 0 -1 0 0 -1 0 0 1 0 -1 1 0 0 -1 0 1 0 0 1 1 0 0 1 1 0 -1 2 -2 1] [ 0 0 0 0 -1 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 -2 1] [ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -2 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -2 0 0 0 1 0 0 1 -1 0 0] [ 0 0 0 0 -1 0 0 0 0 -1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 -1 1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 -1 0 0 1 1 0 0 -1 1 -1] [ 0 -1 0 -1 0 0 0 1 -1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 1 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [ 0 0 0 0 -1 0 0 0 0 -1 0 0 0 -1 0 0 0 0 1 1 0 0 1 1 0 -1 2 -1 1] [ 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1 1 0 0 0 2 0 0 2 0 0 0 2 -2 1] [ 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 -1 -1 1 0 1 1 0 0 -2 1 0 0 0 0 0 1 0 -1 1] [ 0 0 0 0 0 1 -1 0 0 0 -1 -1 1 0 1 1 0 1 -1 1 -1 0 0 0 1 0 1 -1 1] sage: view(T2) sage: charpoly(T2) _38 = x^33 - x^32 - 52*x^31 + 46*x^30 + 1219*x^29 - 939*x4*x^27 + 11220*x^26 + 158471*x^25 - 87155*x^24 - 1034158*016*x^22 + 4872986*x^21 - 1705222*x^20 - 16801171*x^19 + 18 + 42496356*x^17 - 7598744*x^16 - 78351538*x^15 + 81962103678217*x^13 - 3810733*x^12 - 95894495*x^11 - 2530565*x8374*x^9 + 5119626*x^8 - 23144358*x^7 - 3273560*x^6 + 505 943872*x^4 - 502432*x^3 - 103920*x^2 + 16528*x + 3552 sage: factor(charpoly(T2)) _39 = (x - 3) * (x + 2) * (x^2 - 2) * (x^3 - 4*x - 2) * (5 - 2*x^4 - 8*x^3 + 2*x^2 + 4*x - 1) * (x^20 - 3*x^19 - 21*x^17 + 338*x^16 - 1130*x^15 - 2023*x^14 + 7432*x^13 + 6 28021*x^11 - 10909*x^10 + 61267*x^9 + 6954*x^8 - 74752*xx^6 + 46330*x^5 - 1087*x^4 - 12558*x^3 - 942*x^2 + 960*x sage: view(factor(charpoly(T2))) sage: was@form:~/168/notes/2005-09-26$