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TT... ....... ....TT... ........... ........TT... 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1.44391 1.40417 1.36331 1.32135 1.27833 1.23429 1.18927 1.14329 1.09639 1.04861 1.00000 0.95058 0.90041 0.84951 0.79793 0.74571 0.69290 0.63953 0.58565 0.53129 0.47652 0.42136 0.36586 0.31007 0.25404 0.19779 0.14140 0.08488 0.02830 -0.02830 -0.08488 -0.14140 -0.19779 -0.25404 -0.31007 -0.36586 -0.42136 -0.47652 -0.53129 -0.58565 -0.63953 -0.69290 -0.74571 -0.79793 -0.84951 -0.90041 -0.95058 -1.00000 -1.04861 -1.09639 -1.14329 -1.18927 -1.23429 -1.27833 -1.32135 -1.36331 -1.40417 -1.44391 -1.48250 -1.51989 -1.55607 -1.59100 -1.62466 -1.65702 -1.68805 -1.71773 -1.74603 -1.77293 -1.79841 -1.82246 -1.84504 -1.86614 -1.88575 -1.90385 -1.92043 -1.93547 -1.94895 -1.96088 -1.97123 -1.98001 -1.98720 -1.99279 -1.99680 -1.99920 -2.00000 -1.99920 -1.99680 -1.99279 -1.98720 -1.98001 -1.97123 -1.96088 -1.94895 -1.93547 -1.92043 -1.90385 -1.88575 -1.86614 -1.84504 -1.82246 -1.79841 -1.77293 -1.74603 -1.71773 -1.68805 -1.65702 -1.62466 -1.59100 -1.55607 -1.51989 -1.48250 -1.44391 -1.40417 -1.36331 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TT........... E7 2.00000 1.99891 1.99564 1.99020 1.98258 1.97280 1.96088 1.94681 1.93063 1.91233 1.89196 1.86952 1.84504 1.81855 1.79008 1.75966 1.72731 1.69309 1.65702 1.61914 1.57950 1.53814 1.49509 1.45042 1.40417 1.35639 1.30713 1.25644 1.20438 1.15101 1.09639 1.04057 0.98362 0.92559 0.86655 0.80657 0.74571 0.68404 0.62162 0.55853 0.49482 0.43058 0.36586 0.30075 0.23531 0.16961 0.10373 0.03773 -0.02830 -0.09431 -0.16021 -0.22594 -0.29142 -0.35658 -0.42136 -0.48567 -0.54946 -0.61265 -0.67517 -0.73695 -0.79793 -0.85804 -0.91722 -0.97539 -1.03250 -1.08849 -1.14329 -1.19684 -1.24909 -1.29997 -1.34944 -1.39744 -1.44391 -1.48881 -1.53209 -1.57370 -1.61359 -1.65172 -1.68805 -1.72254 -1.75515 -1.78585 -1.81460 -1.84138 -1.86614 -1.88888 -1.90955 -1.92814 -1.94463 -1.95900 -1.97123 -1.98132 -1.98924 -1.99500 -1.99858 -1.99998 -1.99920 -1.99624 -1.99111 -1.98380 -1.97433 -1.96271 -1.94895 -1.93307 -1.91507 -1.89499 -1.87285 -1.84866 -1.82246 -1.79427 -1.76412 -1.73205 -1.69809 -1.66228 -1.62466 -1.58527 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0.05660 -0.00943 -0.07546 -0.14140 -0.20718 -0.27274 -0.33800 -0.40290 -0.46735 -0.53129 -0.59466 -0.65738 -0.71938 -0.78059 -0.84096 -0.90041 -0.95887 -1.01630 -1.07261 -1.12775 -1.18167 -1.23429 -1.28558 -1.33545 -1.38388 -1.43079 -1.47615 -1.51989 -1.56198 -1.60237 -1.64101 -1.67785 -1.71288 -1.74603 -1.77728 -1.80659 -1.83393 -1.85928 -1.88259 -1.90385 -1.92304 -1.94013 -1.95511 -1.96796 -1.97865 -1.98720 -1.99357 -1.99778 -1.99980 -1.99964 -1.99731 -1.99279 -1.98611 -1.97726 -1.96625 -1.95310 -1.93782 -1.92043 -1.90094 -1.87939 -1.85578 -1.83015 -1.80252 -1.77293 -1.74141 -1.70799 -1.67270 -1.63559 -1.59670 -1.55607 -1.51374 -1.46977 -1.42418 -1.37705 -1.32842 -1.27833 -1.22686 -1.17404 -1.11995 -1.06463 -1.00816 -0.95058 -0.89197 -0.83239 -0.77190 -0.71057 -0.64846 -0.58565 -0.52219 -0.45817 -0.39365 -0.32870 -0.26339 -0.19779 -0.13198 -0.06603 0.00000 0.06603 0.13198 0.19779 0.26339 0.32870 0.39365 0.45817 0.52219 0.58565 0.64846 0.71057 0.77190 0.83239 0.89197 0.95058 1.00816 1.06463 1.11995 1.17404 1.22686 1.27833 1.32842 1.37705 1.42418 1.46977 1.51374 1.55607 1.59670 1.63559 1.67270 1.70799 1.74141 1.77293 1.80252 1.83015 1.85578 1.87939 1.90094 1.92043 1.93782 1.95310 1.96625 1.97726 1.98611 1.99279 1.99731 1.99964 1.99980 1.99778 1.99357 1.98720 1.97865 1.96796 1.95511 1.94013 1.92304 1.90385 1.88259 1.85928 1.83393 1.80659 1.77728 1.74603 1.71288 1.67785 1.64101 1.60237 1.56198 1.51989 1.47615 1.43079 1.38388 1.33545 1.28558 1.23429 1.18167 1.12775 1.07261 1.01630 0.95887 0.90041 0.84096 0.78059 0.71938 0.65738 0.59466 0.53129 0.46735 0.40290 0.33800 0.27274 0.20718 0.14140 0.07546 0.00943 -0.05660 -0.12257 -0.18840 -0.25404 -0.31939 -0.38440 -0.44898 -0.51308 -0.57662 -0.63953 -0.70174 -0.76319 -0.82380 -0.88352 -0.94227 -1.00000 -1.05664 -1.11212 -1.16639 -1.21939 -1.27106 -1.32135 -1.37019 -1.41755 -1.46335 -1.50756 -1.55013 -1.59100 -1.63015 -1.66751 -1.70306 -1.73675 -1.76855 -1.79841 -1.82632 -1.85224 -1.87614 -1.89799 -1.91777 -1.93547 -1.95105 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1.68805 1.65172 1.61359 1.57370 1.53209 1.48881 1.44391 1.39744 1.34944 1.29997 1.24909 1.19684 1.14329 1.08849 1.03250 0.97539 0.91722 0.85804 0.79793 0.73695 0.67517 0.61265 0.54946 0.48567 0.42136 0.35658 0.29142 0.22594 0.16021 0.09431 0.02830 -0.03773 -0.10373 -0.16961 -0.23531 -0.30075 -0.36586 -0.43058 -0.49482 -0.55853 -0.62162 -0.68404 -0.74571 -0.80657 -0.86655 -0.92559 -0.98362 -1.04057 -1.09639 -1.15101 -1.20438 -1.25644 -1.30713 -1.35639 -1.40417 -1.45042 -1.49509 -1.53814 -1.57950 -1.61914 -1.65702 -1.69309 -1.72731 -1.75966 -1.79008 -1.81855 -1.84504 -1.86952 -1.89196 -1.91233 -1.93063 -1.94681 -1.96088 -1.97280 -1.98258 -1.99020 -1.99564 -1.99891 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.99858 1.99431 1.98720 1.97726 1.96450 1.94895 1.93063 1.90955 1.88575 1.85928 1.83015 1.79841 1.76412 1.72731 1.68805 1.64638 1.60237 1.55607 1.50756 1.45690 1.40417 1.34944 1.29279 1.23429 1.17404 1.11212 1.04861 0.98362 0.91722 0.84951 0.78059 0.71057 0.63953 0.56758 0.49482 0.42136 0.34730 0.27274 0.19779 0.12257 0.04717 -0.02830 -0.10373 -0.17901 -0.25404 -0.32870 -0.40290 -0.47652 -0.54946 -0.62162 -0.69290 -0.76319 -0.83239 -0.90041 -0.96714 -1.03250 -1.09639 -1.15872 -1.21939 -1.27833 -1.33545 -1.39067 -1.44391 -1.49509 -1.54415 -1.59100 -1.63559 -1.67785 -1.71773 -1.75515 -1.79008 -1.82246 -1.85224 -1.87939 -1.90385 -1.92561 -1.94463 -1.96088 -1.97433 -1.98498 -1.99279 -1.99778 -1.99991 -1.99920 -1.99564 -1.98924 -1.98001 -1.96796 -1.95310 -1.93547 -1.91507 -1.89196 -1.86614 -1.83768 -1.80659 -1.77293 -1.73675 -1.69809 -1.65702 -1.61359 -1.56786 -1.51989 -1.46977 -1.41755 -1.36331 -1.30713 -1.24909 -1.18927 -1.12775 -1.06463 -1.00000 -0.93394 -0.86655 -0.79793 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E501 2.00000 -1.42418 0.02830 1.38388 -1.99920 1.46335 -0.08488 -1.34246 1.99680 -1.50134 0.14140 1.29997 -1.99279 1.53814 -0.19779 -1.25644 1.98720 -1.57370 0.25404 1.21190 -1.98001 1.60799 -0.31007 -1.16639 1.97123 -1.64101 0.36586 1.11995 -1.96088 1.67270 -0.42136 -1.07261 1.94895 -1.70306 0.47652 1.02441 -1.93547 1.73205 -0.53129 -0.97539 1.92043 -1.75966 0.58565 0.92559 -1.90385 1.78585 -0.63953 -0.87505 1.88575 -1.81062 0.69290 0.82380 -1.86614 1.83393 -0.74571 -0.77190 1.84504 -1.85578 0.79793 0.71938 -1.82246 1.87614 -0.84951 -0.66628 1.79841 -1.89499 0.90041 0.61265 -1.77293 1.91233 -0.95058 -0.55853 1.74603 -1.92814 1.00000 0.50396 -1.71773 1.94240 -1.04861 -0.44898 1.68805 -1.95511 1.09639 0.39365 -1.65702 1.96625 -1.14329 -0.33800 1.62466 -1.97582 1.18927 0.28208 -1.59100 1.98380 -1.23429 -0.22594 1.55607 -1.99020 1.27833 0.16961 -1.51989 1.99500 -1.32135 -0.11315 1.48250 -1.99820 1.36331 0.05660 -1.44391 1.99980 -1.40417 0.00000 1.40417 -1.99980 1.44391 -0.05660 -1.36331 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E665 2.00000 -1.99998 1.99991 -1.99980 1.99964 -1.99944 1.99920 -1.99891 1.99858 -1.99820 1.99778 -1.99731 1.99680 -1.99624 1.99564 -1.99500 1.99431 -1.99357 1.99279 -1.99197 1.99111 -1.99020 1.98924 -1.98824 1.98720 -1.98611 1.98498 -1.98380 1.98258 -1.98132 1.98001 -1.97865 1.97726 -1.97582 1.97433 -1.97280 1.97123 -1.96962 1.96796 -1.96625 1.96450 -1.96271 1.96088 -1.95900 1.95708 -1.95511 1.95310 -1.95105 1.94895 -1.94681 1.94463 -1.94240 1.94013 -1.93782 1.93547 -1.93307 1.93063 -1.92814 1.92561 -1.92304 1.92043 -1.91777 1.91507 -1.91233 1.90955 -1.90672 1.90385 -1.90094 1.89799 -1.89499 1.89196 -1.88888 1.88575 -1.88259 1.87939 -1.87614 1.87285 -1.86952 1.86614 -1.86273 1.85928 -1.85578 1.85224 -1.84866 1.84504 -1.84138 1.83768 -1.83393 1.83015 -1.82632 1.82246 -1.81855 1.81460 -1.81062 1.80659 -1.80252 1.79841 -1.79427 1.79008 -1.78585 1.78158 -1.77728 1.77293 -1.76855 1.76412 -1.75966 1.75515 -1.75061 1.74603 -1.74141 1.73675 -1.73205 1.72731 -1.72254 1.71773 -1.71288 1.70799 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1.85578 -1.85928 1.86273 -1.86614 1.86952 -1.87285 1.87614 -1.87939 1.88259 -1.88575 1.88888 -1.89196 1.89499 -1.89799 1.90094 -1.90385 1.90672 -1.90955 1.91233 -1.91507 1.91777 -1.92043 1.92304 -1.92561 1.92814 -1.93063 1.93307 -1.93547 1.93782 -1.94013 1.94240 -1.94463 1.94681 -1.94895 1.95105 -1.95310 1.95511 -1.95708 1.95900 -1.96088 1.96271 -1.96450 1.96625 -1.96796 1.96962 -1.97123 1.97280 -1.97433 1.97582 -1.97726 1.97865 -1.98001 1.98132 -1.98258 1.98380 -1.98498 1.98611 -1.98720 1.98824 -1.98924 1.99020 -1.99111 1.99197 -1.99279 1.99357 -1.99431 1.99500 -1.99564 1.99624 -1.99680 1.99731 -1.99778 1.99820 -1.99858 1.99891 -1.99920 1.99944 -1.99964 1.99980 -1.99991 1.99998 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999977748951 = 2*cos(2*π/1332) = 2*cos(π/666) 1.999910996301 = 2*cos(4*π/1332) = 2*cos(π/333) 1.999799743533 = 2*cos(6*π/1332) = 2*cos(π/222) 1.999643993125 = 2*cos(8*π/1332) = 2*cos(2*π/333) 1.999443748541 = 2*cos(10*π/1332) = 2*cos(5*π/666) 1.999199014237 = 2*cos(12*π/1332) = 2*cos(π/111) 1.998909795658 = 2*cos(14*π/1332) = 2*cos(7*π/666) 1.998576099241 = 2*cos(16*π/1332) = 2*cos(4*π/333) 1.998197932409 = 2*cos(18*π/1332) = 2*cos(π/74) 1.997775303579 = 2*cos(20*π/1332) = 2*cos(5*π/333) 1.997308222153 = 2*cos(22*π/1332) = 2*cos(11*π/666) 1.996796698525 = 2*cos(24*π/1332) = 2*cos(2*π/111) 1.996240744076 = 2*cos(26*π/1332) = 2*cos(13*π/666) 1.995640371178 = 2*cos(28*π/1332) = 2*cos(7*π/333) 1.994995593189 = 2*cos(30*π/1332) = 2*cos(5*π/222) 1.994306424456 = 2*cos(32*π/1332) = 2*cos(8*π/333) 1.993572880314 = 2*cos(34*π/1332) = 2*cos(17*π/666) 1.992794977085 = 2*cos(36*π/1332) = 2*cos(π/37) 1.991972732078 = 2*cos(38*π/1332) = 2*cos(19*π/666) 1.991106163589 = 2*cos(40*π/1332) = 2*cos(10*π/333) 1.990195290901 = 2*cos(42*π/1332) = 2*cos(7*π/222) 1.989240134280 = 2*cos(44*π/1332) = 2*cos(11*π/333) 1.988240714980 = 2*cos(46*π/1332) = 2*cos(23*π/666) 1.987197055239 = 2*cos(48*π/1332) = 2*cos(4*π/111) 1.986109178281 = 2*cos(50*π/1332) = 2*cos(25*π/666) 1.984977108310 = 2*cos(52*π/1332) = 2*cos(13*π/333) 1.983800870518 = 2*cos(54*π/1332) = 2*cos(3*π/74) 1.982580491076 = 2*cos(56*π/1332) = 2*cos(14*π/333) 1.981315997139 = 2*cos(58*π/1332) = 2*cos(29*π/666) 1.980007416844 = 2*cos(60*π/1332) = 2*cos(5*π/111) 1.978654779307 = 2*cos(62*π/1332) = 2*cos(31*π/666) 1.977258114627 = 2*cos(64*π/1332) = 2*cos(16*π/333) 1.975817453880 = 2*cos(66*π/1332) = 2*cos(11*π/222) 1.974332829124 = 2*cos(68*π/1332) = 2*cos(17*π/333) 1.972804273391 = 2*cos(70*π/1332) = 2*cos(35*π/666) 1.971231820695 = 2*cos(72*π/1332) = 2*cos(2*π/37) 1.969615506024 = 2*cos(74*π/1332) = 2*cos(π/18) 1.967955365343 = 2*cos(76*π/1332) = 2*cos(19*π/333) 1.966251435591 = 2*cos(78*π/1332) = 2*cos(13*π/222) 1.964503754683 = 2*cos(80*π/1332) = 2*cos(20*π/333) 1.962712361507 = 2*cos(82*π/1332) = 2*cos(41*π/666) 1.960877295923 = 2*cos(84*π/1332) = 2*cos(7*π/111) 1.958998598762 = 2*cos(86*π/1332) = 2*cos(43*π/666) 1.957076311829 = 2*cos(88*π/1332) = 2*cos(22*π/333) 1.955110477895 = 2*cos(90*π/1332) = 2*cos(5*π/74) 1.953101140704 = 2*cos(92*π/1332) = 2*cos(23*π/333) 1.951048344964 = 2*cos(94*π/1332) = 2*cos(47*π/666) 1.948952136352 = 2*cos(96*π/1332) = 2*cos(8*π/111) 1.946812561512 = 2*cos(98*π/1332) = 2*cos(49*π/666) 1.944629668051 = 2*cos(100*π/1332) = 2*cos(25*π/333) 1.942403504541 = 2*cos(102*π/1332) = 2*cos(17*π/222) 1.940134120516 = 2*cos(104*π/1332) = 2*cos(26*π/333) 1.937821566472 = 2*cos(106*π/1332) = 2*cos(53*π/666) 1.935465893867 = 2*cos(108*π/1332) = 2*cos(3*π/37) 1.933067155116 = 2*cos(110*π/1332) = 2*cos(55*π/666) 1.930625403594 = 2*cos(112*π/1332) = 2*cos(28*π/333) 1.928140693632 = 2*cos(114*π/1332) = 2*cos(19*π/222) 1.925613080518 = 2*cos(116*π/1332) = 2*cos(29*π/333) 1.923042620494 = 2*cos(118*π/1332) = 2*cos(59*π/666) 1.920429370755 = 2*cos(120*π/1332) = 2*cos(10*π/111) 1.917773389449 = 2*cos(122*π/1332) = 2*cos(61*π/666) 1.915074735674 = 2*cos(124*π/1332) = 2*cos(31*π/333) 1.912333469479 = 2*cos(126*π/1332) = 2*cos(7*π/74) 1.909549651858 = 2*cos(128*π/1332) = 2*cos(32*π/333) 1.906723344755 = 2*cos(130*π/1332) = 2*cos(65*π/666) 1.903854611058 = 2*cos(132*π/1332) = 2*cos(11*π/111) 1.900943514600 = 2*cos(134*π/1332) = 2*cos(67*π/666) 1.897990120156 = 2*cos(136*π/1332) = 2*cos(34*π/333) 1.894994493441 = 2*cos(138*π/1332) = 2*cos(23*π/222) 1.891956701111 = 2*cos(140*π/1332) = 2*cos(35*π/333) 1.888876810762 = 2*cos(142*π/1332) = 2*cos(71*π/666) 1.885754890922 = 2*cos(144*π/1332) = 2*cos(4*π/37) 1.882591011059 = 2*cos(146*π/1332) = 2*cos(73*π/666) 1.879385241572 = 2*cos(148*π/1332) = 2*cos(π/9) 1.876137653792 = 2*cos(150*π/1332) = 2*cos(25*π/222) 1.872848319983 = 2*cos(152*π/1332) = 2*cos(38*π/333) 1.869517313334 = 2*cos(154*π/1332) = 2*cos(77*π/666) 1.866144707965 = 2*cos(156*π/1332) = 2*cos(13*π/111) 1.862730578920 = 2*cos(158*π/1332) = 2*cos(79*π/666) 1.859275002166 = 2*cos(160*π/1332) = 2*cos(40*π/333) 1.855778054593 = 2*cos(162*π/1332) = 2*cos(9*π/74) 1.852239814013 = 2*cos(164*π/1332) = 2*cos(41*π/333) 1.848660359155 = 2*cos(166*π/1332) = 2*cos(83*π/666) 1.845039769665 = 2*cos(168*π/1332) = 2*cos(14*π/111) 1.841378126106 = 2*cos(170*π/1332) = 2*cos(85*π/666) 1.837675509952 = 2*cos(172*π/1332) = 2*cos(43*π/333) 1.833932003592 = 2*cos(174*π/1332) = 2*cos(29*π/222) 1.830147690322 = 2*cos(176*π/1332) = 2*cos(44*π/333) 1.826322654346 = 2*cos(178*π/1332) = 2*cos(89*π/666) 1.822456980776 = 2*cos(180*π/1332) = 2*cos(5*π/37) 1.818550755628 = 2*cos(182*π/1332) = 2*cos(91*π/666) 1.814604065818 = 2*cos(184*π/1332) = 2*cos(46*π/333) 1.810616999165 = 2*cos(186*π/1332) = 2*cos(31*π/222) 1.806589644386 = 2*cos(188*π/1332) = 2*cos(47*π/333) 1.802522091092 = 2*cos(190*π/1332) = 2*cos(95*π/666) 1.798414429792 = 2*cos(192*π/1332) = 2*cos(16*π/111) 1.794266751885 = 2*cos(194*π/1332) = 2*cos(97*π/666) 1.790079149661 = 2*cos(196*π/1332) = 2*cos(49*π/333) 1.785851716299 = 2*cos(198*π/1332) = 2*cos(11*π/74) 1.781584545864 = 2*cos(200*π/1332) = 2*cos(50*π/333) 1.777277733305 = 2*cos(202*π/1332) = 2*cos(101*π/666) 1.772931374452 = 2*cos(204*π/1332) = 2*cos(17*π/111) 1.768545566018 = 2*cos(206*π/1332) = 2*cos(103*π/666) 1.764120405590 = 2*cos(208*π/1332) = 2*cos(52*π/333) 1.759655991633 = 2*cos(210*π/1332) = 2*cos(35*π/222) 1.755152423485 = 2*cos(212*π/1332) = 2*cos(53*π/333) 1.750609801356 = 2*cos(214*π/1332) = 2*cos(107*π/666) 1.746028226322 = 2*cos(216*π/1332) = 2*cos(6*π/37) 1.741407800330 = 2*cos(218*π/1332) = 2*cos(109*π/666) 1.736748626189 = 2*cos(220*π/1332) = 2*cos(55*π/333) 1.732050807569 = 2*cos(222*π/1332) = 2*cos(π/6) = √3 1.727314449003 = 2*cos(224*π/1332) = 2*cos(56*π/333) 1.722539655878 = 2*cos(226*π/1332) = 2*cos(113*π/666) 1.717726534441 = 2*cos(228*π/1332) = 2*cos(19*π/111) 1.712875191787 = 2*cos(230*π/1332) = 2*cos(115*π/666) 1.707985735863 = 2*cos(232*π/1332) = 2*cos(58*π/333) 1.703058275467 = 2*cos(234*π/1332) = 2*cos(13*π/74) 1.698092920237 = 2*cos(236*π/1332) = 2*cos(59*π/333) 1.693089780660 = 2*cos(238*π/1332) = 2*cos(119*π/666) 1.688048968060 = 2*cos(240*π/1332) = 2*cos(20*π/111) 1.682970594600 = 2*cos(242*π/1332) = 2*cos(121*π/666) 1.677854773280 = 2*cos(244*π/1332) = 2*cos(61*π/333) 1.672701617932 = 2*cos(246*π/1332) = 2*cos(41*π/222) 1.667511243218 = 2*cos(248*π/1332) = 2*cos(62*π/333) 1.662283764631 = 2*cos(250*π/1332) = 2*cos(125*π/666) 1.657019298488 = 2*cos(252*π/1332) = 2*cos(7*π/37) 1.651717961927 = 2*cos(254*π/1332) = 2*cos(127*π/666) 1.646379872910 = 2*cos(256*π/1332) = 2*cos(64*π/333) 1.641005150214 = 2*cos(258*π/1332) = 2*cos(43*π/222) 1.635593913433 = 2*cos(260*π/1332) = 2*cos(65*π/333) 1.630146282973 = 2*cos(262*π/1332) = 2*cos(131*π/666) 1.624662380048 = 2*cos(264*π/1332) = 2*cos(22*π/111) 1.619142326681 = 2*cos(266*π/1332) = 2*cos(133*π/666) 1.613586245701 = 2*cos(268*π/1332) = 2*cos(67*π/333) 1.607994260734 = 2*cos(270*π/1332) = 2*cos(15*π/74) 1.602366496209 = 2*cos(272*π/1332) = 2*cos(68*π/333) 1.596703077349 = 2*cos(274*π/1332) = 2*cos(137*π/666) 1.591004130171 = 2*cos(276*π/1332) = 2*cos(23*π/111) 1.585269781483 = 2*cos(278*π/1332) = 2*cos(139*π/666) 1.579500158881 = 2*cos(280*π/1332) = 2*cos(70*π/333) 1.573695390743 = 2*cos(282*π/1332) = 2*cos(47*π/222) 1.567855606233 = 2*cos(284*π/1332) = 2*cos(71*π/333) 1.561980935292 = 2*cos(286*π/1332) = 2*cos(143*π/666) 1.556071508637 = 2*cos(288*π/1332) = 2*cos(8*π/37) 1.550127457759 = 2*cos(290*π/1332) = 2*cos(145*π/666) 1.544148914920 = 2*cos(292*π/1332) = 2*cos(73*π/333) 1.538136013149 = 2*cos(294*π/1332) = 2*cos(49*π/222) 1.532088886238 = 2*cos(296*π/1332) = 2*cos(2*π/9) 1.526007668743 = 2*cos(298*π/1332) = 2*cos(149*π/666) 1.519892495977 = 2*cos(300*π/1332) = 2*cos(25*π/111) 1.513743504010 = 2*cos(302*π/1332) = 2*cos(151*π/666) 1.507560829662 = 2*cos(304*π/1332) = 2*cos(76*π/333) 1.501344610505 = 2*cos(306*π/1332) = 2*cos(17*π/74) 1.495094984857 = 2*cos(308*π/1332) = 2*cos(77*π/333) 1.488812091777 = 2*cos(310*π/1332) = 2*cos(155*π/666) 1.482496071067 = 2*cos(312*π/1332) = 2*cos(26*π/111) 1.476147063265 = 2*cos(314*π/1332) = 2*cos(157*π/666) 1.469765209643 = 2*cos(316*π/1332) = 2*cos(79*π/333) 1.463350652203 = 2*cos(318*π/1332) = 2*cos(53*π/222) 1.456903533678 = 2*cos(320*π/1332) = 2*cos(80*π/333) 1.450423997521 = 2*cos(322*π/1332) = 2*cos(161*π/666) 1.443912187909 = 2*cos(324*π/1332) = 2*cos(9*π/37) 1.437368249737 = 2*cos(326*π/1332) = 2*cos(163*π/666) 1.430792328614 = 2*cos(328*π/1332) = 2*cos(82*π/333) 1.424184570862 = 2*cos(330*π/1332) = 2*cos(55*π/222) 1.417545123510 = 2*cos(332*π/1332) = 2*cos(83*π/333) 1.410874134292 = 2*cos(334*π/1332) = 2*cos(167*π/666) 1.404171751645 = 2*cos(336*π/1332) = 2*cos(28*π/111) 1.397438124705 = 2*cos(338*π/1332) = 2*cos(169*π/666) 1.390673403301 = 2*cos(340*π/1332) = 2*cos(85*π/333) 1.383877737955 = 2*cos(342*π/1332) = 2*cos(19*π/74) 1.377051279879 = 2*cos(344*π/1332) = 2*cos(86*π/333) 1.370194180968 = 2*cos(346*π/1332) = 2*cos(173*π/666) 1.363306593799 = 2*cos(348*π/1332) = 2*cos(29*π/111) 1.356388671629 = 2*cos(350*π/1332) = 2*cos(175*π/666) 1.349440568389 = 2*cos(352*π/1332) = 2*cos(88*π/333) 1.342462438682 = 2*cos(354*π/1332) = 2*cos(59*π/222) 1.335454437777 = 2*cos(356*π/1332) = 2*cos(89*π/333) 1.328416721611 = 2*cos(358*π/1332) = 2*cos(179*π/666) 1.321349446780 = 2*cos(360*π/1332) = 2*cos(10*π/37) 1.314252770538 = 2*cos(362*π/1332) = 2*cos(181*π/666) 1.307126850794 = 2*cos(364*π/1332) = 2*cos(91*π/333) 1.299971846107 = 2*cos(366*π/1332) = 2*cos(61*π/222) 1.292787915684 = 2*cos(368*π/1332) = 2*cos(92*π/333) 1.285575219373 = 2*cos(370*π/1332) = 2*cos(5*π/18) 1.278333917666 = 2*cos(372*π/1332) = 2*cos(31*π/111) 1.271064171689 = 2*cos(374*π/1332) = 2*cos(187*π/666) 1.263766143201 = 2*cos(376*π/1332) = 2*cos(94*π/333) 1.256439994591 = 2*cos(378*π/1332) = 2*cos(21*π/74) 1.249085888874 = 2*cos(380*π/1332) = 2*cos(95*π/333) 1.241703989686 = 2*cos(382*π/1332) = 2*cos(191*π/666) 1.234294461283 = 2*cos(384*π/1332) = 2*cos(32*π/111) 1.226857468533 = 2*cos(386*π/1332) = 2*cos(193*π/666) 1.219393176919 = 2*cos(388*π/1332) = 2*cos(97*π/333) 1.211901752527 = 2*cos(390*π/1332) = 2*cos(65*π/222) 1.204383362051 = 2*cos(392*π/1332) = 2*cos(98*π/333) 1.196838172782 = 2*cos(394*π/1332) = 2*cos(197*π/666) 1.189266352609 = 2*cos(396*π/1332) = 2*cos(11*π/37) 1.181668070012 = 2*cos(398*π/1332) = 2*cos(199*π/666) 1.174043494062 = 2*cos(400*π/1332) = 2*cos(100*π/333) 1.166392794413 = 2*cos(402*π/1332) = 2*cos(67*π/222) 1.158716141301 = 2*cos(404*π/1332) = 2*cos(101*π/333) 1.151013705540 = 2*cos(406*π/1332) = 2*cos(203*π/666) 1.143285658517 = 2*cos(408*π/1332) = 2*cos(34*π/111) 1.135532172189 = 2*cos(410*π/1332) = 2*cos(205*π/666) 1.127753419080 = 2*cos(412*π/1332) = 2*cos(103*π/333) 1.119949572275 = 2*cos(414*π/1332) = 2*cos(23*π/74) 1.112120805418 = 2*cos(416*π/1332) = 2*cos(104*π/333) 1.104267292706 = 2*cos(418*π/1332) = 2*cos(209*π/666) 1.096389208889 = 2*cos(420*π/1332) = 2*cos(35*π/111) 1.088486729263 = 2*cos(422*π/1332) = 2*cos(211*π/666) 1.080560029666 = 2*cos(424*π/1332) = 2*cos(106*π/333) 1.072609286475 = 2*cos(426*π/1332) = 2*cos(71*π/222) 1.064634676602 = 2*cos(428*π/1332) = 2*cos(107*π/333) 1.056636377492 = 2*cos(430*π/1332) = 2*cos(215*π/666) 1.048614567114 = 2*cos(432*π/1332) = 2*cos(12*π/37) 1.040569423963 = 2*cos(434*π/1332) = 2*cos(217*π/666) 1.032501127051 = 2*cos(436*π/1332) = 2*cos(109*π/333) 1.024409855906 = 2*cos(438*π/1332) = 2*cos(73*π/222) 1.016295790568 = 2*cos(440*π/1332) = 2*cos(110*π/333) 1.008159111583 = 2*cos(442*π/1332) = 2*cos(221*π/666) 0.991818637369 = 2*cos(446*π/1332) = 2*cos(223*π/666) 0.983615205733 = 2*cos(448*π/1332) = 2*cos(112*π/333) 0.975389887627 = 2*cos(450*π/1332) = 2*cos(25*π/74) 0.967142866074 = 2*cos(452*π/1332) = 2*cos(113*π/333) 0.958874324578 = 2*cos(454*π/1332) = 2*cos(227*π/666) 0.950584447122 = 2*cos(456*π/1332) = 2*cos(38*π/111) 0.942273418166 = 2*cos(458*π/1332) = 2*cos(229*π/666) 0.933941422638 = 2*cos(460*π/1332) = 2*cos(115*π/333) 0.925588645935 = 2*cos(462*π/1332) = 2*cos(77*π/222) 0.917215273913 = 2*cos(464*π/1332) = 2*cos(116*π/333) 0.908821492890 = 2*cos(466*π/1332) = 2*cos(233*π/666) 0.900407489635 = 2*cos(468*π/1332) = 2*cos(13*π/37) 0.891973451370 = 2*cos(470*π/1332) = 2*cos(235*π/666) 0.883519565760 = 2*cos(472*π/1332) = 2*cos(118*π/333) 0.875046020914 = 2*cos(474*π/1332) = 2*cos(79*π/222) 0.866553005376 = 2*cos(476*π/1332) = 2*cos(119*π/333) 0.858040708125 = 2*cos(478*π/1332) = 2*cos(239*π/666) 0.849509318568 = 2*cos(480*π/1332) = 2*cos(40*π/111) 0.840959026538 = 2*cos(482*π/1332) = 2*cos(241*π/666) 0.832390022289 = 2*cos(484*π/1332) = 2*cos(121*π/333) 0.823802496488 = 2*cos(486*π/1332) = 2*cos(27*π/74) 0.815196640218 = 2*cos(488*π/1332) = 2*cos(122*π/333) 0.806572644968 = 2*cos(490*π/1332) = 2*cos(245*π/666) 0.797930702631 = 2*cos(492*π/1332) = 2*cos(41*π/111) 0.789271005499 = 2*cos(494*π/1332) = 2*cos(247*π/666) 0.780593746259 = 2*cos(496*π/1332) = 2*cos(124*π/333) 0.771899117991 = 2*cos(498*π/1332) = 2*cos(83*π/222) 0.763187314157 = 2*cos(500*π/1332) = 2*cos(125*π/333) 0.754458528606 = 2*cos(502*π/1332) = 2*cos(251*π/666) 0.745712955561 = 2*cos(504*π/1332) = 2*cos(14*π/37) 0.736950789621 = 2*cos(506*π/1332) = 2*cos(253*π/666) 0.728172225753 = 2*cos(508*π/1332) = 2*cos(127*π/333) 0.719377459289 = 2*cos(510*π/1332) = 2*cos(85*π/222) 0.710566685923 = 2*cos(512*π/1332) = 2*cos(128*π/333) 0.701740101703 = 2*cos(514*π/1332) = 2*cos(257*π/666) 0.692897903029 = 2*cos(516*π/1332) = 2*cos(43*π/111) 0.684040286651 = 2*cos(518*π/1332) = 2*cos(7*π/18) 0.675167449660 = 2*cos(520*π/1332) = 2*cos(130*π/333) 0.666279589484 = 2*cos(522*π/1332) = 2*cos(29*π/74) 0.657376903889 = 2*cos(524*π/1332) = 2*cos(131*π/333) 0.648459590969 = 2*cos(526*π/1332) = 2*cos(263*π/666) 0.639527849142 = 2*cos(528*π/1332) = 2*cos(44*π/111) 0.630581877151 = 2*cos(530*π/1332) = 2*cos(265*π/666) 0.621621874052 = 2*cos(532*π/1332) = 2*cos(133*π/333) 0.612648039214 = 2*cos(534*π/1332) = 2*cos(89*π/222) 0.603660572314 = 2*cos(536*π/1332) = 2*cos(134*π/333) 0.594659673334 = 2*cos(538*π/1332) = 2*cos(269*π/666) 0.585645542553 = 2*cos(540*π/1332) = 2*cos(15*π/37) 0.576618380544 = 2*cos(542*π/1332) = 2*cos(271*π/666) 0.567578388172 = 2*cos(544*π/1332) = 2*cos(136*π/333) 0.558525766586 = 2*cos(546*π/1332) = 2*cos(91*π/222) 0.549460717215 = 2*cos(548*π/1332) = 2*cos(137*π/333) 0.540383441768 = 2*cos(550*π/1332) = 2*cos(275*π/666) 0.531294142222 = 2*cos(552*π/1332) = 2*cos(46*π/111) 0.522193020824 = 2*cos(554*π/1332) = 2*cos(277*π/666) 0.513080280084 = 2*cos(556*π/1332) = 2*cos(139*π/333) 0.503956122770 = 2*cos(558*π/1332) = 2*cos(31*π/74) 0.494820751904 = 2*cos(560*π/1332) = 2*cos(140*π/333) 0.485674370757 = 2*cos(562*π/1332) = 2*cos(281*π/666) 0.476517182846 = 2*cos(564*π/1332) = 2*cos(47*π/111) 0.467349391929 = 2*cos(566*π/1332) = 2*cos(283*π/666) 0.458171201997 = 2*cos(568*π/1332) = 2*cos(142*π/333) 0.448982817275 = 2*cos(570*π/1332) = 2*cos(95*π/222) 0.439784442215 = 2*cos(572*π/1332) = 2*cos(143*π/333) 0.430576281490 = 2*cos(574*π/1332) = 2*cos(287*π/666) 0.421358539991 = 2*cos(576*π/1332) = 2*cos(16*π/37) 0.412131422823 = 2*cos(578*π/1332) = 2*cos(289*π/666) 0.402895135299 = 2*cos(580*π/1332) = 2*cos(145*π/333) 0.393649882935 = 2*cos(582*π/1332) = 2*cos(97*π/222) 0.384395871449 = 2*cos(584*π/1332) = 2*cos(146*π/333) 0.375133306752 = 2*cos(586*π/1332) = 2*cos(293*π/666) 0.365862394945 = 2*cos(588*π/1332) = 2*cos(49*π/111) 0.356583342316 = 2*cos(590*π/1332) = 2*cos(295*π/666) 0.347296355334 = 2*cos(592*π/1332) = 2*cos(4*π/9) 0.338001640644 = 2*cos(594*π/1332) = 2*cos(33*π/74) 0.328699405063 = 2*cos(596*π/1332) = 2*cos(149*π/333) 0.319389855575 = 2*cos(598*π/1332) = 2*cos(299*π/666) 0.310073199328 = 2*cos(600*π/1332) = 2*cos(50*π/111) 0.300749643628 = 2*cos(602*π/1332) = 2*cos(301*π/666) 0.291419395932 = 2*cos(604*π/1332) = 2*cos(151*π/333) 0.282082663850 = 2*cos(606*π/1332) = 2*cos(101*π/222) 0.272739655132 = 2*cos(608*π/1332) = 2*cos(152*π/333) 0.263390577671 = 2*cos(610*π/1332) = 2*cos(305*π/666) 0.254035639494 = 2*cos(612*π/1332) = 2*cos(17*π/37) 0.244675048757 = 2*cos(614*π/1332) = 2*cos(307*π/666) 0.235309013744 = 2*cos(616*π/1332) = 2*cos(154*π/333) 0.225937742858 = 2*cos(618*π/1332) = 2*cos(103*π/222) 0.216561444621 = 2*cos(620*π/1332) = 2*cos(155*π/333) 0.207180327664 = 2*cos(622*π/1332) = 2*cos(311*π/666) 0.197794600728 = 2*cos(624*π/1332) = 2*cos(52*π/111) 0.188404472655 = 2*cos(626*π/1332) = 2*cos(313*π/666) 0.179010152385 = 2*cos(628*π/1332) = 2*cos(157*π/333) 0.169611848951 = 2*cos(630*π/1332) = 2*cos(35*π/74) 0.160209771476 = 2*cos(632*π/1332) = 2*cos(158*π/333) 0.150804129165 = 2*cos(634*π/1332) = 2*cos(317*π/666) 0.141395131304 = 2*cos(636*π/1332) = 2*cos(53*π/111) 0.131982987253 = 2*cos(638*π/1332) = 2*cos(319*π/666) 0.122567906443 = 2*cos(640*π/1332) = 2*cos(160*π/333) 0.113150098368 = 2*cos(642*π/1332) = 2*cos(107*π/222) 0.103729772584 = 2*cos(644*π/1332) = 2*cos(161*π/333) 0.094307138705 = 2*cos(646*π/1332) = 2*cos(323*π/666) 0.084882406392 = 2*cos(648*π/1332) = 2*cos(18*π/37) 0.075455785357 = 2*cos(650*π/1332) = 2*cos(325*π/666) 0.066027485352 = 2*cos(652*π/1332) = 2*cos(163*π/333) 0.056597716166 = 2*cos(654*π/1332) = 2*cos(109*π/222) 0.047166687622 = 2*cos(656*π/1332) = 2*cos(164*π/333) 0.037734609569 = 2*cos(658*π/1332) = 2*cos(329*π/666) 0.028301691882 = 2*cos(660*π/1332) = 2*cos(55*π/111) 0.018868144452 = 2*cos(662*π/1332) = 2*cos(331*π/666) 0.009434177186 = 2*cos(664*π/1332) = 2*cos(166*π/333) ******* 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 0.000 0.000 −0.000 0.000 −0.000 0.000 0.000 0.000 −0.000 −0.000 −0.000 0.000 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−0.000 −0.000 0.000 −0.000 −0.000 −0.000 0.000 −0.000 −0.000 0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 −0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 0.000 0.000 −0.000 −0.000 −0.000 −0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 −0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 −0.000 0.000 0.000 0.000 *************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************** Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R Rz B2 p+f+h+j+l z, z3, z5 E1 p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2 d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3 f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4 g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+l E8 k+m E9 l E10 m E657 m E658 l E659 k+m E660 j+l E661 i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E662 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E663 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E664 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E665 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D666d point group is 2664, and the order of the principal axis (S1332) is 1332. The group has 669 irreducible representations. β The D666d point group is isomorphic to C1332v and D1332. γ The D666d point group is generated by two symmetry elements, S1332 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C2′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D666d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. ι The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.
| D340d | ||
| D512d | ||
| C666 C666v C666h D666 D666h | D666d | S666 |
This Character Table for the D666d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.