C82 E 2 C82 2 C41 2 C82^3 2 C41^2 2 C82^5 2 C41^3 2 C82^7 2 C41^4 2 C82^9 2 C41^5 2 C82^11 2 C41^6 2 C82^13 2 C41^7 2 C82^15 2 C41^8 2 C82^17 2 C41^9 2 C82^19 2 C41^10 2 C82^21 2 C41^11 2 C82^23 2 C41^12 2 C82^25 2 C41^13 2 C82^27 2 C41^14 2 C82^29 2 C41^15 2 C82^31 2 C41^16 2 C82^33 2 C41^17 2 C82^35 2 C41^18 2 C82^37 2 C41^19 2 C82^39 2 C41^20 C2 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ..T ..T ....T ......T ........T ..........T ............T B 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1 * 2.00000 1.99413 1.97656 1.94739 1.90679 1.85500 1.79233 1.71914 1.63586 1.54298 1.44104 1.33065 1.21245 1.08714 0.95544 0.81814 0.67603 0.52996 0.38078 0.22937 0.07661 -0.07661 -0.22937 -0.38078 -0.52996 -0.67603 -0.81814 -0.95544 -1.08714 -1.21245 -1.33065 -1.44104 -1.54298 -1.63586 -1.71914 -1.79233 -1.85500 -1.90679 -1.94739 -1.97656 -1.99413 -2.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.97656 1.90679 1.79233 1.63586 1.44104 1.21245 0.95544 0.67603 0.38078 0.07661 -0.22937 -0.52996 -0.81814 -1.08714 -1.33065 -1.54298 -1.71914 -1.85500 -1.94739 -1.99413 -1.99413 -1.94739 -1.85500 -1.71914 -1.54298 -1.33065 -1.08714 -0.81814 -0.52996 -0.22937 0.07661 0.38078 0.67603 0.95544 1.21245 1.44104 1.63586 1.79233 1.90679 1.97656 2.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.94739 1.79233 1.54298 1.21245 0.81814 0.38078 -0.07661 -0.52996 -0.95544 -1.33065 -1.63586 -1.85500 -1.97656 -1.99413 -1.90679 -1.71914 -1.44104 -1.08714 -0.67603 -0.22937 0.22937 0.67603 1.08714 1.44104 1.71914 1.90679 1.99413 1.97656 1.85500 1.63586 1.33065 0.95544 0.52996 0.07661 -0.38078 -0.81814 -1.21245 -1.54298 -1.79233 -1.94739 -2.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 2.00000 1.90679 1.63586 1.21245 0.67603 0.07661 -0.52996 -1.08714 -1.54298 -1.85500 -1.99413 -1.94739 -1.71914 -1.33065 -0.81814 -0.22937 0.38078 0.95544 1.44104 1.79233 1.97656 1.97656 1.79233 1.44104 0.95544 0.38078 -0.22937 -0.81814 -1.33065 -1.71914 -1.94739 -1.99413 -1.85500 -1.54298 -1.08714 -0.52996 0.07661 0.67603 1.21245 1.63586 1.90679 2.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.00000 1.85500 1.44104 0.81814 0.07661 -0.67603 -1.33065 -1.79233 -1.99413 -1.90679 -1.54298 -0.95544 -0.22937 0.52996 1.21245 1.71914 1.97656 1.94739 1.63586 1.08714 0.38078 -0.38078 -1.08714 -1.63586 -1.94739 -1.97656 -1.71914 -1.21245 -0.52996 0.22937 0.95544 1.54298 1.90679 1.99413 1.79233 1.33065 0.67603 -0.07661 -0.81814 -1.44104 -1.85500 -2.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.00000 1.79233 1.21245 0.38078 -0.52996 -1.33065 -1.85500 -1.99413 -1.71914 -1.08714 -0.22937 0.67603 1.44104 1.90679 1.97656 1.63586 0.95544 0.07661 -0.81814 -1.54298 -1.94739 -1.94739 -1.54298 -0.81814 0.07661 0.95544 1.63586 1.97656 1.90679 1.44104 0.67603 -0.22937 -1.08714 -1.71914 -1.99413 -1.85500 -1.33065 -0.52996 0.38078 1.21245 1.79233 2.00000 ... ... ..... ....... ......... ........... TT........... E7 * 2.00000 1.71914 0.95544 -0.07661 -1.08714 -1.79233 -1.99413 -1.63586 -0.81814 0.22937 1.21245 1.85500 1.97656 1.54298 0.67603 -0.38078 -1.33065 -1.90679 -1.94739 -1.44104 -0.52996 0.52996 1.44104 1.94739 1.90679 1.33065 0.38078 -0.67603 -1.54298 -1.97656 -1.85500 -1.21245 -0.22937 0.81814 1.63586 1.99413 1.79233 1.08714 0.07661 -0.95544 -1.71914 -2.00000 ... ... ..... ....... ......... ........... ............. E8 * 2.00000 1.63586 0.67603 -0.52996 -1.54298 -1.99413 -1.71914 -0.81814 0.38078 1.44104 1.97656 1.79233 0.95544 -0.22937 -1.33065 -1.94739 -1.85500 -1.08714 0.07661 1.21245 1.90679 1.90679 1.21245 0.07661 -1.08714 -1.85500 -1.94739 -1.33065 -0.22937 0.95544 1.79233 1.97656 1.44104 0.38078 -0.81814 -1.71914 -1.99413 -1.54298 -0.52996 0.67603 1.63586 2.00000 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 1.54298 0.38078 -0.95544 -1.85500 -1.90679 -1.08714 0.22937 1.44104 1.99413 1.63586 0.52996 -0.81814 -1.79233 -1.94739 -1.21245 0.07661 1.33065 1.97656 1.71914 0.67603 -0.67603 -1.71914 -1.97656 -1.33065 -0.07661 1.21245 1.94739 1.79233 0.81814 -0.52996 -1.63586 -1.99413 -1.44104 -0.22937 1.08714 1.90679 1.85500 0.95544 -0.38078 -1.54298 -2.00000 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 1.44104 0.07661 -1.33065 -1.99413 -1.54298 -0.22937 1.21245 1.97656 1.63586 0.38078 -1.08714 -1.94739 -1.71914 -0.52996 0.95544 1.90679 1.79233 0.67603 -0.81814 -1.85500 -1.85500 -0.81814 0.67603 1.79233 1.90679 0.95544 -0.52996 -1.71914 -1.94739 -1.08714 0.38078 1.63586 1.97656 1.21245 -0.22937 -1.54298 -1.99413 -1.33065 0.07661 1.44104 2.00000 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 1.33065 -0.22937 -1.63586 -1.94739 -0.95544 0.67603 1.85500 1.79233 0.52996 -1.08714 -1.97656 -1.54298 -0.07661 1.44104 1.99413 1.21245 -0.38078 -1.71914 -1.90679 -0.81814 0.81814 1.90679 1.71914 0.38078 -1.21245 -1.99413 -1.44104 0.07661 1.54298 1.97656 1.08714 -0.52996 -1.79233 -1.85500 -0.67603 0.95544 1.94739 1.63586 0.22937 -1.33065 -2.00000 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 1.21245 -0.52996 -1.85500 -1.71914 -0.22937 1.44104 1.97656 0.95544 -0.81814 -1.94739 -1.54298 0.07661 1.63586 1.90679 0.67603 -1.08714 -1.99413 -1.33065 0.38078 1.79233 1.79233 0.38078 -1.33065 -1.99413 -1.08714 0.67603 1.90679 1.63586 0.07661 -1.54298 -1.94739 -0.81814 0.95544 1.97656 1.44104 -0.22937 -1.71914 -1.85500 -0.52996 1.21245 2.00000 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 1.08714 -0.81814 -1.97656 -1.33065 0.52996 1.90679 1.54298 -0.22937 -1.79233 -1.71914 -0.07661 1.63586 1.85500 0.38078 -1.44104 -1.94739 -0.67603 1.21245 1.99413 0.95544 -0.95544 -1.99413 -1.21245 0.67603 1.94739 1.44104 -0.38078 -1.85500 -1.63586 0.07661 1.71914 1.79233 0.22937 -1.54298 -1.90679 -0.52996 1.33065 1.97656 0.81814 -1.08714 -2.00000 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 0.95544 -1.08714 -1.99413 -0.81814 1.21245 1.97656 0.67603 -1.33065 -1.94739 -0.52996 1.44104 1.90679 0.38078 -1.54298 -1.85500 -0.22937 1.63586 1.79233 0.07661 -1.71914 -1.71914 0.07661 1.79233 1.63586 -0.22937 -1.85500 -1.54298 0.38078 1.90679 1.44104 -0.52996 -1.94739 -1.33065 0.67603 1.97656 1.21245 -0.81814 -1.99413 -1.08714 0.95544 2.00000 ... ... ..... ....... ......... ........... ............. E15 * 2.00000 0.81814 -1.33065 -1.90679 -0.22937 1.71914 1.63586 -0.38078 -1.94739 -1.21245 0.95544 1.99413 0.67603 -1.44104 -1.85500 -0.07661 1.79233 1.54298 -0.52996 -1.97656 -1.08714 1.08714 1.97656 0.52996 -1.54298 -1.79233 0.07661 1.85500 1.44104 -0.67603 -1.99413 -0.95544 1.21245 1.94739 0.38078 -1.63586 -1.71914 0.22937 1.90679 1.33065 -0.81814 -2.00000 ... ... ..... ....... ......... ........... ............. E16 * 2.00000 0.67603 -1.54298 -1.71914 0.38078 1.97656 0.95544 -1.33065 -1.85500 0.07661 1.90679 1.21245 -1.08714 -1.94739 -0.22937 1.79233 1.44104 -0.81814 -1.99413 -0.52996 1.63586 1.63586 -0.52996 -1.99413 -0.81814 1.44104 1.79233 -0.22937 -1.94739 -1.08714 1.21245 1.90679 0.07661 -1.85500 -1.33065 0.95544 1.97656 0.38078 -1.71914 -1.54298 0.67603 2.00000 ... ... ..... ....... ......... ........... ............. E17 * 2.00000 0.52996 -1.71914 -1.44104 0.95544 1.94739 0.07661 -1.90679 -1.08714 1.33065 1.79233 -0.38078 -1.99413 -0.67603 1.63586 1.54298 -0.81814 -1.97656 -0.22937 1.85500 1.21245 -1.21245 -1.85500 0.22937 1.97656 0.81814 -1.54298 -1.63586 0.67603 1.99413 0.38078 -1.79233 -1.33065 1.08714 1.90679 -0.07661 -1.94739 -0.95544 1.44104 1.71914 -0.52996 -2.00000 ... ... ..... ....... ......... ........... ............. E18 * 2.00000 0.38078 -1.85500 -1.08714 1.44104 1.63586 -0.81814 -1.94739 0.07661 1.97656 0.67603 -1.71914 -1.33065 1.21245 1.79233 -0.52996 -1.99413 -0.22937 1.90679 0.95544 -1.54298 -1.54298 0.95544 1.90679 -0.22937 -1.99413 -0.52996 1.79233 1.21245 -1.33065 -1.71914 0.67603 1.97656 0.07661 -1.94739 -0.81814 1.63586 1.44104 -1.08714 -1.85500 0.38078 2.00000 ... ... ..... ....... ......... ........... ............. E19 * 2.00000 0.22937 -1.94739 -0.67603 1.79233 1.08714 -1.54298 -1.44104 1.21245 1.71914 -0.81814 -1.90679 0.38078 1.99413 0.07661 -1.97656 -0.52996 1.85500 0.95544 -1.63586 -1.33065 1.33065 1.63586 -0.95544 -1.85500 0.52996 1.97656 -0.07661 -1.99413 -0.38078 1.90679 0.81814 -1.71914 -1.21245 1.44104 1.54298 -1.08714 -1.79233 0.67603 1.94739 -0.22937 -2.00000 ... ... ..... ....... ......... ........... ............. E20 * 2.00000 0.07661 -1.99413 -0.22937 1.97656 0.38078 -1.94739 -0.52996 1.90679 0.67603 -1.85500 -0.81814 1.79233 0.95544 -1.71914 -1.08714 1.63586 1.21245 -1.54298 -1.33065 1.44104 1.44104 -1.33065 -1.54298 1.21245 1.63586 -1.08714 -1.71914 0.95544 1.79233 -0.81814 -1.85500 0.67603 1.90679 -0.52996 -1.94739 0.38078 1.97656 -0.22937 -1.99413 0.07661 2.00000 ... ... ..... ....... ......... ........... ............. E21 * 2.00000 -0.07661 -1.99413 0.22937 1.97656 -0.38078 -1.94739 0.52996 1.90679 -0.67603 -1.85500 0.81814 1.79233 -0.95544 -1.71914 1.08714 1.63586 -1.21245 -1.54298 1.33065 1.44104 -1.44104 -1.33065 1.54298 1.21245 -1.63586 -1.08714 1.71914 0.95544 -1.79233 -0.81814 1.85500 0.67603 -1.90679 -0.52996 1.94739 0.38078 -1.97656 -0.22937 1.99413 0.07661 -2.00000 ... ... ..... ....... ......... ........... ............. E22 * 2.00000 -0.22937 -1.94739 0.67603 1.79233 -1.08714 -1.54298 1.44104 1.21245 -1.71914 -0.81814 1.90679 0.38078 -1.99413 0.07661 1.97656 -0.52996 -1.85500 0.95544 1.63586 -1.33065 -1.33065 1.63586 0.95544 -1.85500 -0.52996 1.97656 0.07661 -1.99413 0.38078 1.90679 -0.81814 -1.71914 1.21245 1.44104 -1.54298 -1.08714 1.79233 0.67603 -1.94739 -0.22937 2.00000 ... ... ..... ....... ......... ........... ............. E23 * 2.00000 -0.38078 -1.85500 1.08714 1.44104 -1.63586 -0.81814 1.94739 0.07661 -1.97656 0.67603 1.71914 -1.33065 -1.21245 1.79233 0.52996 -1.99413 0.22937 1.90679 -0.95544 -1.54298 1.54298 0.95544 -1.90679 -0.22937 1.99413 -0.52996 -1.79233 1.21245 1.33065 -1.71914 -0.67603 1.97656 -0.07661 -1.94739 0.81814 1.63586 -1.44104 -1.08714 1.85500 0.38078 -2.00000 ... ... ..... ....... ......... ........... ............. E24 * 2.00000 -0.52996 -1.71914 1.44104 0.95544 -1.94739 0.07661 1.90679 -1.08714 -1.33065 1.79233 0.38078 -1.99413 0.67603 1.63586 -1.54298 -0.81814 1.97656 -0.22937 -1.85500 1.21245 1.21245 -1.85500 -0.22937 1.97656 -0.81814 -1.54298 1.63586 0.67603 -1.99413 0.38078 1.79233 -1.33065 -1.08714 1.90679 0.07661 -1.94739 0.95544 1.44104 -1.71914 -0.52996 2.00000 ... ... ..... ....... ......... ........... ............. E25 * 2.00000 -0.67603 -1.54298 1.71914 0.38078 -1.97656 0.95544 1.33065 -1.85500 -0.07661 1.90679 -1.21245 -1.08714 1.94739 -0.22937 -1.79233 1.44104 0.81814 -1.99413 0.52996 1.63586 -1.63586 -0.52996 1.99413 -0.81814 -1.44104 1.79233 0.22937 -1.94739 1.08714 1.21245 -1.90679 0.07661 1.85500 -1.33065 -0.95544 1.97656 -0.38078 -1.71914 1.54298 0.67603 -2.00000 ... ... ..... ....... ......... ........... ............. E26 * 2.00000 -0.81814 -1.33065 1.90679 -0.22937 -1.71914 1.63586 0.38078 -1.94739 1.21245 0.95544 -1.99413 0.67603 1.44104 -1.85500 0.07661 1.79233 -1.54298 -0.52996 1.97656 -1.08714 -1.08714 1.97656 -0.52996 -1.54298 1.79233 0.07661 -1.85500 1.44104 0.67603 -1.99413 0.95544 1.21245 -1.94739 0.38078 1.63586 -1.71914 -0.22937 1.90679 -1.33065 -0.81814 2.00000 ... ... ..... ....... ......... ........... ............. E27 * 2.00000 -0.95544 -1.08714 1.99413 -0.81814 -1.21245 1.97656 -0.67603 -1.33065 1.94739 -0.52996 -1.44104 1.90679 -0.38078 -1.54298 1.85500 -0.22937 -1.63586 1.79233 -0.07661 -1.71914 1.71914 0.07661 -1.79233 1.63586 0.22937 -1.85500 1.54298 0.38078 -1.90679 1.44104 0.52996 -1.94739 1.33065 0.67603 -1.97656 1.21245 0.81814 -1.99413 1.08714 0.95544 -2.00000 ... ... ..... ....... ......... ........... ............. E28 * 2.00000 -1.08714 -0.81814 1.97656 -1.33065 -0.52996 1.90679 -1.54298 -0.22937 1.79233 -1.71914 0.07661 1.63586 -1.85500 0.38078 1.44104 -1.94739 0.67603 1.21245 -1.99413 0.95544 0.95544 -1.99413 1.21245 0.67603 -1.94739 1.44104 0.38078 -1.85500 1.63586 0.07661 -1.71914 1.79233 -0.22937 -1.54298 1.90679 -0.52996 -1.33065 1.97656 -0.81814 -1.08714 2.00000 ... ... ..... ....... ......... ........... ............. E29 * 2.00000 -1.21245 -0.52996 1.85500 -1.71914 0.22937 1.44104 -1.97656 0.95544 0.81814 -1.94739 1.54298 0.07661 -1.63586 1.90679 -0.67603 -1.08714 1.99413 -1.33065 -0.38078 1.79233 -1.79233 0.38078 1.33065 -1.99413 1.08714 0.67603 -1.90679 1.63586 -0.07661 -1.54298 1.94739 -0.81814 -0.95544 1.97656 -1.44104 -0.22937 1.71914 -1.85500 0.52996 1.21245 -2.00000 ... ... ..... ....... ......... ........... ............. E30 * 2.00000 -1.33065 -0.22937 1.63586 -1.94739 0.95544 0.67603 -1.85500 1.79233 -0.52996 -1.08714 1.97656 -1.54298 0.07661 1.44104 -1.99413 1.21245 0.38078 -1.71914 1.90679 -0.81814 -0.81814 1.90679 -1.71914 0.38078 1.21245 -1.99413 1.44104 0.07661 -1.54298 1.97656 -1.08714 -0.52996 1.79233 -1.85500 0.67603 0.95544 -1.94739 1.63586 -0.22937 -1.33065 2.00000 ... ... ..... ....... ......... ........... ............. E31 * 2.00000 -1.44104 0.07661 1.33065 -1.99413 1.54298 -0.22937 -1.21245 1.97656 -1.63586 0.38078 1.08714 -1.94739 1.71914 -0.52996 -0.95544 1.90679 -1.79233 0.67603 0.81814 -1.85500 1.85500 -0.81814 -0.67603 1.79233 -1.90679 0.95544 0.52996 -1.71914 1.94739 -1.08714 -0.38078 1.63586 -1.97656 1.21245 0.22937 -1.54298 1.99413 -1.33065 -0.07661 1.44104 -2.00000 ... ... ..... ....... ......... ........... ............. E32 * 2.00000 -1.54298 0.38078 0.95544 -1.85500 1.90679 -1.08714 -0.22937 1.44104 -1.99413 1.63586 -0.52996 -0.81814 1.79233 -1.94739 1.21245 0.07661 -1.33065 1.97656 -1.71914 0.67603 0.67603 -1.71914 1.97656 -1.33065 0.07661 1.21245 -1.94739 1.79233 -0.81814 -0.52996 1.63586 -1.99413 1.44104 -0.22937 -1.08714 1.90679 -1.85500 0.95544 0.38078 -1.54298 2.00000 ... ... ..... ....... ......... ........... ............. E33 * 2.00000 -1.63586 0.67603 0.52996 -1.54298 1.99413 -1.71914 0.81814 0.38078 -1.44104 1.97656 -1.79233 0.95544 0.22937 -1.33065 1.94739 -1.85500 1.08714 0.07661 -1.21245 1.90679 -1.90679 1.21245 -0.07661 -1.08714 1.85500 -1.94739 1.33065 -0.22937 -0.95544 1.79233 -1.97656 1.44104 -0.38078 -0.81814 1.71914 -1.99413 1.54298 -0.52996 -0.67603 1.63586 -2.00000 ... ... ..... ....... ......... ........... ............. E34 * 2.00000 -1.71914 0.95544 0.07661 -1.08714 1.79233 -1.99413 1.63586 -0.81814 -0.22937 1.21245 -1.85500 1.97656 -1.54298 0.67603 0.38078 -1.33065 1.90679 -1.94739 1.44104 -0.52996 -0.52996 1.44104 -1.94739 1.90679 -1.33065 0.38078 0.67603 -1.54298 1.97656 -1.85500 1.21245 -0.22937 -0.81814 1.63586 -1.99413 1.79233 -1.08714 0.07661 0.95544 -1.71914 2.00000 ... ... ..... ....... ......... ........... ............. E35 * 2.00000 -1.79233 1.21245 -0.38078 -0.52996 1.33065 -1.85500 1.99413 -1.71914 1.08714 -0.22937 -0.67603 1.44104 -1.90679 1.97656 -1.63586 0.95544 -0.07661 -0.81814 1.54298 -1.94739 1.94739 -1.54298 0.81814 0.07661 -0.95544 1.63586 -1.97656 1.90679 -1.44104 0.67603 0.22937 -1.08714 1.71914 -1.99413 1.85500 -1.33065 0.52996 0.38078 -1.21245 1.79233 -2.00000 ... ... ..... ....... ......... ........... ............. E36 * 2.00000 -1.85500 1.44104 -0.81814 0.07661 0.67603 -1.33065 1.79233 -1.99413 1.90679 -1.54298 0.95544 -0.22937 -0.52996 1.21245 -1.71914 1.97656 -1.94739 1.63586 -1.08714 0.38078 0.38078 -1.08714 1.63586 -1.94739 1.97656 -1.71914 1.21245 -0.52996 -0.22937 0.95544 -1.54298 1.90679 -1.99413 1.79233 -1.33065 0.67603 0.07661 -0.81814 1.44104 -1.85500 2.00000 ... ... ..... ....... ......... ........... ............. E37 * 2.00000 -1.90679 1.63586 -1.21245 0.67603 -0.07661 -0.52996 1.08714 -1.54298 1.85500 -1.99413 1.94739 -1.71914 1.33065 -0.81814 0.22937 0.38078 -0.95544 1.44104 -1.79233 1.97656 -1.97656 1.79233 -1.44104 0.95544 -0.38078 -0.22937 0.81814 -1.33065 1.71914 -1.94739 1.99413 -1.85500 1.54298 -1.08714 0.52996 0.07661 -0.67603 1.21245 -1.63586 1.90679 -2.00000 ... ... ..... ....... ......... ........... ............. E38 * 2.00000 -1.94739 1.79233 -1.54298 1.21245 -0.81814 0.38078 0.07661 -0.52996 0.95544 -1.33065 1.63586 -1.85500 1.97656 -1.99413 1.90679 -1.71914 1.44104 -1.08714 0.67603 -0.22937 -0.22937 0.67603 -1.08714 1.44104 -1.71914 1.90679 -1.99413 1.97656 -1.85500 1.63586 -1.33065 0.95544 -0.52996 0.07661 0.38078 -0.81814 1.21245 -1.54298 1.79233 -1.94739 2.00000 ... ... ..... ....... ......... ........... ............. E39 * 2.00000 -1.97656 1.90679 -1.79233 1.63586 -1.44104 1.21245 -0.95544 0.67603 -0.38078 0.07661 0.22937 -0.52996 0.81814 -1.08714 1.33065 -1.54298 1.71914 -1.85500 1.94739 -1.99413 1.99413 -1.94739 1.85500 -1.71914 1.54298 -1.33065 1.08714 -0.81814 0.52996 -0.22937 -0.07661 0.38078 -0.67603 0.95544 -1.21245 1.44104 -1.63586 1.79233 -1.90679 1.97656 -2.00000 ... ... ..... ....... ......... ........... ............. E40 * 2.00000 -1.99413 1.97656 -1.94739 1.90679 -1.85500 1.79233 -1.71914 1.63586 -1.54298 1.44104 -1.33065 1.21245 -1.08714 0.95544 -0.81814 0.67603 -0.52996 0.38078 -0.22937 0.07661 0.07661 -0.22937 0.38078 -0.52996 0.67603 -0.81814 0.95544 -1.08714 1.21245 -1.33065 1.44104 -1.54298 1.63586 -1.71914 1.79233 -1.85500 1.90679 -1.94739 1.97656 -1.99413 2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.994131602367 = 2*cos(2*π/82) = 2*cos(π/41) 1.976560847561 = 2*cos(4*π/82) = 2*cos(2*π/41) 1.947390847756 = 2*cos(6*π/82) = 2*cos(3*π/41) 1.906792784110 = 2*cos(8*π/82) = 2*cos(4*π/41) 1.855004902204 = 2*cos(10*π/82) = 2*cos(5*π/41) 1.792331113922 = 2*cos(12*π/82) = 2*cos(6*π/41) 1.719139213974 = 2*cos(14*π/82) = 2*cos(7*π/41) 1.635858721533 = 2*cos(16*π/82) = 2*cos(8*π/41) 1.542978359644 = 2*cos(18*π/82) = 2*cos(9*π/41) 1.441043187202 = 2*cos(20*π/82) = 2*cos(10*π/41) 1.330651400331 = 2*cos(22*π/82) = 2*cos(11*π/41) 1.212450821933 = 2*cos(24*π/82) = 2*cos(12*π/41) 1.087135100002 = 2*cos(26*π/82) = 2*cos(13*π/41) 0.955439637025 = 2*cos(28*π/82) = 2*cos(14*π/41) 0.818137274343 = 2*cos(30*π/82) = 2*cos(15*π/41) 0.676033756817 = 2*cos(32*π/82) = 2*cos(16*π/41) 0.529963004393 = 2*cos(34*π/82) = 2*cos(17*π/41) 0.380782218329 = 2*cos(36*π/82) = 2*cos(18*π/41) 0.229366850797 = 2*cos(38*π/82) = 2*cos(19*π/41) 0.076605467380 = 2*cos(40*π/82) = 2*cos(20*π/41) Symmetry of Rotations and Cartesian products A R+p+d+f+g+h+i+j+k+l+m Rz, z, z2, z3, z4, z5, z6 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the C82 point group is 82, and the order of the principal axis (C82) is 82. The group has 42 irreducible representations. β The C82 point group is isomorphic to S82 and C41h. γ The C82 point group is generated by one single symmetry element, C82. Therefore, it is a cyclic group. δ The lowest nonvanishing multipole moment in C82 is 2 (dipole moment). ε This is an Abelian point group (the commutative law holds between all symmetry operations). The C82 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ζ Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 80 cases have been combined into 40 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 40 pairs of left and right rotations have been combined into one two-membered pseudo-class each. η The 40 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. θ The point group is chiral, as it does not contain any mirroring operation. ι 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. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| C80 | ||
| C81 | ||
| C82 | C82v C82h D82 D82h D82d S82 | |
| C83 | ||
| C84 |
This Character Table for the C82 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.