CoCalc Public Fileswww / 168 / notes / 2005-09-28 / sage-session.txtOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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[email protected]:~/168/notes/2005-09-26$ sage
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SAGE Version 0.7.4, Export Date: 2005-09-27-0000
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Distributed under the terms of the GNU General Public L
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IPython shell -- for help type <object>?, <object>??, %
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sage: M = ModularSymbols(11)
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sage: M
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_2 = Full Modular Symbols space for Gamma_0(11) of weighign 0 and dimension 3 over Rational Field
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sage: M.basis()
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_3 = ((1,0), (1,8), (1,9))
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sage: M = ModularSymbols(11,sign=1)
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sage: M.basis()
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_5 = ((1,0), (1,9))
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sage: M([1,0])
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---------------------------------------------------------
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Traceback (most recent call last):
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File "<console>", line 1, in ?
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File "/home/was/sage/local/lib/python2.4/site-packages/ar/modsym/ambient.py", line 214, in __call__
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raise TypeError, "No coercion of %s into %s defined."
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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.
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sage: M.basis()
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_7 = ((1,0), (1,9))
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sage: M.hecke_matrix(2)
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_8 =
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[ 3 -1]
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[ 0 -2]
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sage: M = ModularSymbols(37,sign=1)
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sage: M.basis()
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_10 = ((1,0), (1,23), (1,34))
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sage: T7 = M.hecke_matrix(7)
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sage: T7
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_12 =
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[ 8 0 -6]
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[ 0 -1 0]
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[ 0 0 -1]
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sage: charpoly(T7)
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_13 = x^3 - 6*x^2 - 15*x - 8
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sage: factor(charpoly(T7))
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_14 = (x - 8) * (x + 1)^2
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sage: M.hecke_matrix(11)
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_15 =
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[12 0 -6]
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[ 0 -5 0]
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[ 0 0 3]
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sage: T13 = M.hecke_matrix(13)
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sage: T7 * T13
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_17 =
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[112 0 -72]
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[ 0 2 0]
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[ 0 0 4]
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sage: T13 * T7
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_18 =
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[112 0 -72]
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[ 0 2 0]
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[ 0 0 4]
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sage: factor(charpoly(T13))
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_19 = (x - 14) * (x + 2) * (x + 4)
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sage: kernel(T13+2)
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_20 =
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Vector space of degree 3 and dimension 1 over Rational Fi
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Basis matrix:
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[0 1 0]
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sage: kernel(T13-14)
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_21 =
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Vector space of degree 3 and dimension 1 over Rational Fi
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Basis matrix:
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[ 1 0 -2/3]
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sage: M.basis()
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_22 = ((1,0), (1,23), (1,34))
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sage: R = Integers(37); R(1/23)
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_23 = 29
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sage: R(1/34)
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_24 = 12
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sage: kernel(T13+4)
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_25 =
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Vector space of degree 3 and dimension 1 over Rational Fi
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Basis matrix:
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[0 0 1]
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sage: factor(charpoly(T13))
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_26 = (x - 14) * (x + 2) * (x + 4)
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sage: v = kernel(T13+4).basis()[0]
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sage: v
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_28 = (0, 0, 1)
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sage: T17 = M.hecke_matrix(17)
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sage: v*T17
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_30 = (0, 0, 6)
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sage: M = ModularSymbols(2,sign=1)
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sage: T13 = M.hecke_matrix(13)
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sage: T13
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_33 = [14]
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sage: M = ModularSymbols(389,sign=1)
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sage: T2 = M.heck
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M.hecke_algebra M.hecke_module_of_level
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M.hecke_bound M.hecke_operator
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M.hecke_matrix M.hecke_polynomial
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sage: T2 = M.hecke_matrix(2)
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sage: T2
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_36 =
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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[ 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]
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sage: view(T2)
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sage: charpoly(T2)
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_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
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sage: factor(charpoly(T2))
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_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
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sage: view(factor(charpoly(T2)))
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sage:
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[email protected]:~/168/notes/2005-09-26$
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