CoCalc Public Fileswww / 168 / notes / 2005-09-28 / sage-session.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1[email protected]:~/168/notes/2005-09-26$sage 2--------------------------------------------------------- 3 SAGE Version 0.7.4, Export Date: 2005-09-27-0000 4 Distributed under the terms of the GNU General Public L 5 IPython shell -- for help type <object>?, <object>??, % 6--------------------------------------------------------- 7 8sage: M = ModularSymbols(11) 9sage: M 10 _2 = Full Modular Symbols space for Gamma_0(11) of weighign 0 and dimension 3 over Rational Field 11sage: M.basis() 12 _3 = ((1,0), (1,8), (1,9)) 13sage: M = ModularSymbols(11,sign=1) 14sage: M.basis() 15 _5 = ((1,0), (1,9)) 16sage: M([1,0]) 17--------------------------------------------------------- 18Traceback (most recent call last): 19 File "<console>", line 1, in ? 20 File "/home/was/sage/local/lib/python2.4/site-packages/ar/modsym/ambient.py", line 214, in __call__ 21 raise TypeError, "No coercion of %s into %s defined." 22TypeError: No coercion of [1, 0] into Full Modular Symbolr Gamma_0(11) of weight 2 with sign 1 and dimension 2 ove Field defined. 23 24sage: M.basis() 25 _7 = ((1,0), (1,9)) 26sage: M.hecke_matrix(2) 27 _8 = 28[ 3 -1] 29[ 0 -2] 30sage: M = ModularSymbols(37,sign=1) 31sage: M.basis() 32_10 = ((1,0), (1,23), (1,34)) 33sage: T7 = M.hecke_matrix(7) 34sage: T7 35_12 = 36[ 8 0 -6] 37[ 0 -1 0] 38[ 0 0 -1] 39sage: charpoly(T7) 40_13 = x^3 - 6*x^2 - 15*x - 8 41sage: factor(charpoly(T7)) 42_14 = (x - 8) * (x + 1)^2 43sage: M.hecke_matrix(11) 44_15 = 45[12 0 -6] 46[ 0 -5 0] 47[ 0 0 3] 48sage: T13 = M.hecke_matrix(13) 49sage: T7 * T13 50_17 = 51[112 0 -72] 52[ 0 2 0] 53[ 0 0 4] 54sage: T13 * T7 55_18 = 56[112 0 -72] 57[ 0 2 0] 58[ 0 0 4] 59sage: factor(charpoly(T13)) 60_19 = (x - 14) * (x + 2) * (x + 4) 61sage: kernel(T13+2) 62_20 = 63Vector space of degree 3 and dimension 1 over Rational Fi 64Basis matrix: 65[0 1 0] 66sage: kernel(T13-14) 67_21 = 68Vector space of degree 3 and dimension 1 over Rational Fi 69Basis matrix: 70[ 1 0 -2/3] 71sage: M.basis() 72_22 = ((1,0), (1,23), (1,34)) 73sage: R = Integers(37); R(1/23) 74_23 = 29 75sage: R(1/34) 76_24 = 12 77sage: kernel(T13+4) 78_25 = 79Vector space of degree 3 and dimension 1 over Rational Fi 80Basis matrix: 81[0 0 1] 82sage: factor(charpoly(T13)) 83_26 = (x - 14) * (x + 2) * (x + 4) 84sage: v = kernel(T13+4).basis()[0] 85sage: v 86_28 = (0, 0, 1) 87sage: T17 = M.hecke_matrix(17) 88sage: v*T17 89_30 = (0, 0, 6) 90sage: M = ModularSymbols(2,sign=1) 91sage: T13 = M.hecke_matrix(13) 92sage: T13 93_33 = [14] 94sage: M = ModularSymbols(389,sign=1) 95sage: T2 = M.heck 96M.hecke_algebra M.hecke_module_of_level 97M.hecke_bound M.hecke_operator 98M.hecke_matrix M.hecke_polynomial 99sage: T2 = M.hecke_matrix(2) 100sage: T2 101_36 = 102[ 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] 103[ 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] 104[ 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] 105[ 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] 106[ 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] 107[ 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] 108[ 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] 109[ 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] 110[ 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] 111[ 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] 112[ 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] 113[ 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] 114[ 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] 115[ 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] 116[ 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] 117[ 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] 118[ 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] 119[ 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] 120[ 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] 121[ 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] 122[ 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] 123[ 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] 124[ 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] 125[ 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] 126[ 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] 127[ 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] 128[ 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] 129[ 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] 130[ 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] 131[ 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] 132[ 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] 133[ 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] 134[ 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] 135sage: view(T2) 136sage: charpoly(T2) 137_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 138sage: factor(charpoly(T2)) 139_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 140sage: view(factor(charpoly(T2))) 141sage: 142[email protected]:~/168/notes/2005-09-26$
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