C41h E 2 C41 2 C41^2 2 C41^3 2 C41^4 2 C41^5 2 C41^6 2 C41^7 2 C41^8 2 C41^9 2 C41^10 2 C41^11 2 C41^12 2 C41^13 2 C41^14 2 C41^15 2 C41^16 2 C41^17 2 C41^18 2 C41^19 2 C41^20 sh 2 S41 2 S41^3 2 S41^5 2 S41^7 2 S41^9 2 S41^11 2 S41^13 2 S41^15 2 S41^17 2 S41^19 2 S41^21 2 S41^23 2 S41^25 2 S41^27 2 S41^29 2 S41^31 2 S41^33 2 S41^35 2 S41^37 2 S41^39 <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 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 ............. E1' * 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 2.00000 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 ... TT. ..... ....TT. ......... ........TT. ............. E1" * 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 -2.00000 -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 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' * 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 2.00000 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 ... ... TT... ....... ....TT... ........... ........TT... E2" * 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 -2.00000 -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 ... ... ..... ..TT... ......... ......TT... ............. E3' * 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 2.00000 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 ... ... ..... TT..... ......... ....TT..... ............. E3" * 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 -2.00000 -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 ... ... ..... ....... ..TT..... ........... ......TT..... E4' * 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 2.00000 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 ... ... ..... ....... TT....... ........... ....TT....... E4" * 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 -2.00000 -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 ... ... ..... ....... ......... ..TT....... ............. E5' * 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 2.00000 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 ... ... ..... ....... ......... TT......... ............. E5" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ..TT......... E6' * 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 2.00000 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 ... ... ..... ....... ......... ........... TT........... E6" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E7' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E7" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E8' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E8" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E9' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E9" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E10' * 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 2.00000 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 ... ... ..... ....... ......... ........... ............. E10" * 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 -2.00000 -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 ... ... ..... ....... ......... ........... ............. E11' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E11" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E12' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E12" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E13' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E13" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E14' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E14" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E15' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E15" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E16' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E16" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E17' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E17" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E18' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E18" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E19' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E19" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. E20' * 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 2.00000 -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 ... ... ..... ....... ......... ........... ............. E20" * 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 -2.00000 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 ... ... ..... ....... ......... ........... ............. 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+d+g+i+k+m Rz, z2, z4, z6 A" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l {z((x2−y2)2−4x2y2), xyz(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)} E5" 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)} E6' i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E6" j+l E7' j+l E7" k+m E8' k+m E8" l E9' l E9" m E10' m Notes: α The order of the C41h point group is 82, and the order of the principal axis (S41) is 82. The group has 42 irreducible representations. β The C41h point group could also be named S41, as it contains the S41 axis as its only symmetry element. Another rare designation is C82i because the S41 axis is identical to a roto-inversion axis of order 82. γ The C41h point group is isomorphic to C82 and S82. δ The C41h point group is generated by one single symmetry element, S41. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C41 and σh. ε The lowest nonvanishing multipole moment in C41h is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C41h 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. η 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. θ 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. 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 just 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.
C39h | ||
C40h | ||
C41 C41v | C41h | D41 D41h D41d |
C42h | ||
C43h |
This Character Table for the C41h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.