C77 E 2 C77 2 C77^2 2 C77^3 2 C77^4 2 C77^5 2 C77^6 2 C11 2 C77^8 2 C77^9 2 C77^10 2 C7 2 C77^12 2 C77^13 2 C11^2 2 C77^15 2 C77^16 2 C77^17 2 C77^18 2 C77^19 2 C77^20 2 C11^3 2 C7^2 2 C77^23 2 C77^24 2 C77^25 2 C77^26 2 C77^27 2 C11^4 2 C77^29 2 C77^30 2 C77^31 2 C77^32 2 C7^3 2 C77^34 2 C11^5 2 C77^36 2 C77^37 2 C77^38 <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 ..T ..T ....T ......T ........T ..........T ............T E1 * 2.00000 1.99335 1.97342 1.94037 1.89441 1.83583 1.76504 1.68251 1.58877 1.48447 1.37028 1.24698 1.11538 0.97635 0.83083 0.67978 0.52420 0.36514 0.20365 0.04080 -0.12232 -0.28463 -0.44504 -0.60249 -0.75593 -0.90434 -1.04674 -1.18216 -1.30972 -1.42856 -1.53790 -1.63700 -1.72521 -1.80194 -1.86667 -1.91899 -1.95853 -1.98504 -1.99834 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.97342 1.89441 1.76504 1.58877 1.37028 1.11538 0.83083 0.52420 0.20365 -0.12232 -0.44504 -0.75593 -1.04674 -1.30972 -1.53790 -1.72521 -1.86667 -1.95853 -1.99834 -1.98504 -1.91899 -1.80194 -1.63700 -1.42856 -1.18216 -0.90434 -0.60249 -0.28463 0.04080 0.36514 0.67978 0.97635 1.24698 1.48447 1.68251 1.83583 1.94037 1.99335 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.94037 1.76504 1.48447 1.11538 0.67978 0.20365 -0.28463 -0.75593 -1.18216 -1.53790 -1.80194 -1.95853 -1.99834 -1.91899 -1.72521 -1.42856 -1.04674 -0.60249 -0.12232 0.36514 0.83083 1.24698 1.58877 1.83583 1.97342 1.99335 1.89441 1.68251 1.37028 0.97635 0.52420 0.04080 -0.44504 -0.90434 -1.30972 -1.63700 -1.86667 -1.98504 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 2.00000 1.89441 1.58877 1.11538 0.52420 -0.12232 -0.75593 -1.30972 -1.72521 -1.95853 -1.98504 -1.80194 -1.42856 -0.90434 -0.28463 0.36514 0.97635 1.48447 1.83583 1.99335 1.94037 1.68251 1.24698 0.67978 0.04080 -0.60249 -1.18216 -1.63700 -1.91899 -1.99834 -1.86667 -1.53790 -1.04674 -0.44504 0.20365 0.83083 1.37028 1.76504 1.97342 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.00000 1.83583 1.37028 0.67978 -0.12232 -0.90434 -1.53790 -1.91899 -1.98504 -1.72521 -1.18216 -0.44504 0.36514 1.11538 1.68251 1.97342 1.94037 1.58877 0.97635 0.20365 -0.60249 -1.30972 -1.80194 -1.99834 -1.86667 -1.42856 -0.75593 0.04080 0.83083 1.48447 1.89441 1.99335 1.76504 1.24698 0.52420 -0.28463 -1.04674 -1.63700 -1.95853 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.00000 1.76504 1.11538 0.20365 -0.75593 -1.53790 -1.95853 -1.91899 -1.42856 -0.60249 0.36514 1.24698 1.83583 1.99335 1.68251 0.97635 0.04080 -0.90434 -1.63700 -1.98504 -1.86667 -1.30972 -0.44504 0.52420 1.37028 1.89441 1.97342 1.58877 0.83083 -0.12232 -1.04674 -1.72521 -1.99834 -1.80194 -1.18216 -0.28463 0.67978 1.48447 1.94037 ... ... ..... ....... ......... ........... TT........... E7 * 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 ... ... ..... ....... ......... ........... ............. E8 * 2.00000 1.58877 0.52420 -0.75593 -1.72521 -1.98504 -1.42856 -0.28463 0.97635 1.83583 1.94037 1.24698 0.04080 -1.18216 -1.91899 -1.86667 -1.04674 0.20365 1.37028 1.97342 1.76504 0.83083 -0.44504 -1.53790 -1.99834 -1.63700 -0.60249 0.67978 1.68251 1.99335 1.48447 0.36514 -0.90434 -1.80194 -1.95853 -1.30972 -0.12232 1.11538 1.89441 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 1.48447 0.20365 -1.18216 -1.95853 -1.72521 -0.60249 0.83083 1.83583 1.89441 0.97635 -0.44504 -1.63700 -1.98504 -1.30972 0.04080 1.37028 1.99335 1.58877 0.36514 -1.04674 -1.91899 -1.80194 -0.75593 0.67978 1.76504 1.94037 1.11538 -0.28463 -1.53790 -1.99834 -1.42856 -0.12232 1.24698 1.97342 1.68251 0.52420 -0.90434 -1.86667 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 1.37028 -0.12232 -1.53790 -1.98504 -1.18216 0.36514 1.68251 1.94037 0.97635 -0.60249 -1.80194 -1.86667 -0.75593 0.83083 1.89441 1.76504 0.52420 -1.04674 -1.95853 -1.63700 -0.28463 1.24698 1.99335 1.48447 0.04080 -1.42856 -1.99834 -1.30972 0.20365 1.58877 1.97342 1.11538 -0.44504 -1.72521 -1.91899 -0.90434 0.67978 1.83583 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 1.11538 -0.75593 -1.95853 -1.42856 0.36514 1.83583 1.68251 0.04080 -1.63700 -1.86667 -0.44504 1.37028 1.97342 0.83083 -1.04674 -1.99834 -1.18216 0.67978 1.94037 1.48447 -0.28463 -1.80194 -1.72521 -0.12232 1.58877 1.89441 0.52420 -1.30972 -1.98504 -0.90434 0.97635 1.99335 1.24698 -0.60249 -1.91899 -1.53790 0.20365 1.76504 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 0.97635 -1.04674 -1.99834 -0.90434 1.11538 1.99335 0.83083 -1.18216 -1.98504 -0.75593 1.24698 1.97342 0.67978 -1.30972 -1.95853 -0.60249 1.37028 1.94037 0.52420 -1.42856 -1.91899 -0.44504 1.48447 1.89441 0.36514 -1.53790 -1.86667 -0.28463 1.58877 1.83583 0.20365 -1.63700 -1.80194 -0.12232 1.68251 1.76504 0.04080 -1.72521 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 ... ... ..... ....... ......... ........... ............. E15 * 2.00000 0.67978 -1.53790 -1.72521 0.36514 1.97342 0.97635 -1.30972 -1.86667 0.04080 1.89441 1.24698 -1.04674 -1.95853 -0.28463 1.76504 1.48447 -0.75593 -1.99834 -0.60249 1.58877 1.68251 -0.44504 -1.98504 -0.90434 1.37028 1.83583 -0.12232 -1.91899 -1.18216 1.11538 1.94037 0.20365 -1.80194 -1.42856 0.83083 1.99335 0.52420 -1.63700 ... ... ..... ....... ......... ........... ............. E16 * 2.00000 0.52420 -1.72521 -1.42856 0.97635 1.94037 0.04080 -1.91899 -1.04674 1.37028 1.76504 -0.44504 -1.99834 -0.60249 1.68251 1.48447 -0.90434 -1.95853 -0.12232 1.89441 1.11538 -1.30972 -1.80194 0.36514 1.99335 0.67978 -1.63700 -1.53790 0.83083 1.97342 0.20365 -1.86667 -1.18216 1.24698 1.83583 -0.28463 -1.98504 -0.75593 1.58877 ... ... ..... ....... ......... ........... ............. E17 * 2.00000 0.36514 -1.86667 -1.04674 1.48447 1.58877 -0.90434 -1.91899 0.20365 1.99335 0.52420 -1.80194 -1.18216 1.37028 1.68251 -0.75593 -1.95853 0.04080 1.97342 0.67978 -1.72521 -1.30972 1.24698 1.76504 -0.60249 -1.98504 -0.12232 1.94037 0.83083 -1.63700 -1.42856 1.11538 1.83583 -0.44504 -1.99834 -0.28463 1.89441 0.97635 -1.53790 ... ... ..... ....... ......... ........... ............. E18 * 2.00000 0.20365 -1.95853 -0.60249 1.83583 0.97635 -1.63700 -1.30972 1.37028 1.58877 -1.04674 -1.80194 0.67978 1.94037 -0.28463 -1.99834 -0.12232 1.97342 0.52420 -1.86667 -0.90434 1.68251 1.24698 -1.42856 -1.53790 1.11538 1.76504 -0.75593 -1.91899 0.36514 1.99335 0.04080 -1.98504 -0.44504 1.89441 0.83083 -1.72521 -1.18216 1.48447 ... ... ..... ....... ......... ........... ............. E19 * 2.00000 0.04080 -1.99834 -0.12232 1.99335 0.20365 -1.98504 -0.28463 1.97342 0.36514 -1.95853 -0.44504 1.94037 0.52420 -1.91899 -0.60249 1.89441 0.67978 -1.86667 -0.75593 1.83583 0.83083 -1.80194 -0.90434 1.76504 0.97635 -1.72521 -1.04674 1.68251 1.11538 -1.63700 -1.18216 1.58877 1.24698 -1.53790 -1.30972 1.48447 1.37028 -1.42856 ... ... ..... ....... ......... ........... ............. E20 * 2.00000 -0.12232 -1.98504 0.36514 1.94037 -0.60249 -1.86667 0.83083 1.76504 -1.04674 -1.63700 1.24698 1.48447 -1.42856 -1.30972 1.58877 1.11538 -1.72521 -0.90434 1.83583 0.67978 -1.91899 -0.44504 1.97342 0.20365 -1.99834 0.04080 1.99335 -0.28463 -1.95853 0.52420 1.89441 -0.75593 -1.80194 0.97635 1.68251 -1.18216 -1.53790 1.37028 ... ... ..... ....... ......... ........... ............. E21 * 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 ... ... ..... ....... ......... ........... ............. E22 * 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 ... ... ..... ....... ......... ........... ............. E23 * 2.00000 -0.60249 -1.63700 1.58877 0.67978 -1.99834 0.52420 1.68251 -1.53790 -0.75593 1.99335 -0.44504 -1.72521 1.48447 0.83083 -1.98504 0.36514 1.76504 -1.42856 -0.90434 1.97342 -0.28463 -1.80194 1.37028 0.97635 -1.95853 0.20365 1.83583 -1.30972 -1.04674 1.94037 -0.12232 -1.86667 1.24698 1.11538 -1.91899 0.04080 1.89441 -1.18216 ... ... ..... ....... ......... ........... ............. E24 * 2.00000 -0.75593 -1.42856 1.83583 0.04080 -1.86667 1.37028 0.83083 -1.99834 0.67978 1.48447 -1.80194 -0.12232 1.89441 -1.30972 -0.90434 1.99335 -0.60249 -1.53790 1.76504 0.20365 -1.91899 1.24698 0.97635 -1.98504 0.52420 1.58877 -1.72521 -0.28463 1.94037 -1.18216 -1.04674 1.97342 -0.44504 -1.63700 1.68251 0.36514 -1.95853 1.11538 ... ... ..... ....... ......... ........... ............. E25 * 2.00000 -0.90434 -1.18216 1.97342 -0.60249 -1.42856 1.89441 -0.28463 -1.63700 1.76504 0.04080 -1.80194 1.58877 0.36514 -1.91899 1.37028 0.67978 -1.98504 1.11538 0.97635 -1.99834 0.83083 1.24698 -1.95853 0.52420 1.48447 -1.86667 0.20365 1.68251 -1.72521 -0.12232 1.83583 -1.53790 -0.44504 1.94037 -1.30972 -0.75593 1.99335 -1.04674 ... ... ..... ....... ......... ........... ............. E26 * 2.00000 -1.04674 -0.90434 1.99335 -1.18216 -0.75593 1.97342 -1.30972 -0.60249 1.94037 -1.42856 -0.44504 1.89441 -1.53790 -0.28463 1.83583 -1.63700 -0.12232 1.76504 -1.72521 0.04080 1.68251 -1.80194 0.20365 1.58877 -1.86667 0.36514 1.48447 -1.91899 0.52420 1.37028 -1.95853 0.67978 1.24698 -1.98504 0.83083 1.11538 -1.99834 0.97635 ... ... ..... ....... ......... ........... ............. E27 * 2.00000 -1.18216 -0.60249 1.89441 -1.63700 0.04080 1.58877 -1.91899 0.67978 1.11538 -1.99834 1.24698 0.52420 -1.86667 1.68251 -0.12232 -1.53790 1.94037 -0.75593 -1.04674 1.99335 -1.30972 -0.44504 1.83583 -1.72521 0.20365 1.48447 -1.95853 0.83083 0.97635 -1.98504 1.37028 0.36514 -1.80194 1.76504 -0.28463 -1.42856 1.97342 -0.90434 ... ... ..... ....... ......... ........... ............. E28 * 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 ... ... ..... ....... ......... ........... ............. E29 * 2.00000 -1.42856 0.04080 1.37028 -1.99834 1.48447 -0.12232 -1.30972 1.99335 -1.53790 0.20365 1.24698 -1.98504 1.58877 -0.28463 -1.18216 1.97342 -1.63700 0.36514 1.11538 -1.95853 1.68251 -0.44504 -1.04674 1.94037 -1.72521 0.52420 0.97635 -1.91899 1.76504 -0.60249 -0.90434 1.89441 -1.80194 0.67978 0.83083 -1.86667 1.83583 -0.75593 ... ... ..... ....... ......... ........... ............. E30 * 2.00000 -1.53790 0.36514 0.97635 -1.86667 1.89441 -1.04674 -0.28463 1.48447 -1.99834 1.58877 -0.44504 -0.90434 1.83583 -1.91899 1.11538 0.20365 -1.42856 1.99335 -1.63700 0.52420 0.83083 -1.80194 1.94037 -1.18216 -0.12232 1.37028 -1.98504 1.68251 -0.60249 -0.75593 1.76504 -1.95853 1.24698 0.04080 -1.30972 1.97342 -1.72521 0.67978 ... ... ..... ....... ......... ........... ............. E31 * 2.00000 -1.63700 0.67978 0.52420 -1.53790 1.99335 -1.72521 0.83083 0.36514 -1.42856 1.97342 -1.80194 0.97635 0.20365 -1.30972 1.94037 -1.86667 1.11538 0.04080 -1.18216 1.89441 -1.91899 1.24698 -0.12232 -1.04674 1.83583 -1.95853 1.37028 -0.28463 -0.90434 1.76504 -1.98504 1.48447 -0.44504 -0.75593 1.68251 -1.99834 1.58877 -0.60249 ... ... ..... ....... ......... ........... ............. E32 * 2.00000 -1.72521 0.97635 0.04080 -1.04674 1.76504 -1.99834 1.68251 -0.90434 -0.12232 1.11538 -1.80194 1.99335 -1.63700 0.83083 0.20365 -1.18216 1.83583 -1.98504 1.58877 -0.75593 -0.28463 1.24698 -1.86667 1.97342 -1.53790 0.67978 0.36514 -1.30972 1.89441 -1.95853 1.48447 -0.60249 -0.44504 1.37028 -1.91899 1.94037 -1.42856 0.52420 ... ... ..... ....... ......... ........... ............. E33 * 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 ... ... ..... ....... ......... ........... ............. E34 * 2.00000 -1.86667 1.48447 -0.90434 0.20365 0.52420 -1.18216 1.68251 -1.95853 1.97342 -1.72521 1.24698 -0.60249 -0.12232 0.83083 -1.42856 1.83583 -1.99834 1.89441 -1.53790 0.97635 -0.28463 -0.44504 1.11538 -1.63700 1.94037 -1.98504 1.76504 -1.30972 0.67978 0.04080 -0.75593 1.37028 -1.80194 1.99335 -1.91899 1.58877 -1.04674 0.36514 ... ... ..... ....... ......... ........... ............. E35 * 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 ... ... ..... ....... ......... ........... ............. E36 * 2.00000 -1.95853 1.83583 -1.63700 1.37028 -1.04674 0.67978 -0.28463 -0.12232 0.52420 -0.90434 1.24698 -1.53790 1.76504 -1.91899 1.99335 -1.98504 1.89441 -1.72521 1.48447 -1.18216 0.83083 -0.44504 0.04080 0.36514 -0.75593 1.11538 -1.42856 1.68251 -1.86667 1.97342 -1.99834 1.94037 -1.80194 1.58877 -1.30972 0.97635 -0.60249 0.20365 ... ... ..... ....... ......... ........... ............. E37 * 2.00000 -1.98504 1.94037 -1.86667 1.76504 -1.63700 1.48447 -1.30972 1.11538 -0.90434 0.67978 -0.44504 0.20365 0.04080 -0.28463 0.52420 -0.75593 0.97635 -1.18216 1.37028 -1.53790 1.68251 -1.80194 1.89441 -1.95853 1.99335 -1.99834 1.97342 -1.91899 1.83583 -1.72521 1.58877 -1.42856 1.24698 -1.04674 0.83083 -0.60249 0.36514 -0.12232 ... ... ..... ....... ......... ........... ............. E38 * 2.00000 -1.99834 1.99335 -1.98504 1.97342 -1.95853 1.94037 -1.91899 1.89441 -1.86667 1.83583 -1.80194 1.76504 -1.72521 1.68251 -1.63700 1.58877 -1.53790 1.48447 -1.42856 1.37028 -1.30972 1.24698 -1.18216 1.11538 -1.04674 0.97635 -0.90434 0.83083 -0.75593 0.67978 -0.60249 0.52420 -0.44504 0.36514 -0.28463 0.20365 -0.12232 0.04080 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.993345164988 = 2*cos(2*π/77) 1.973424946783 = 2*cos(4*π/77) 1.940371911148 = 2*cos(6*π/77) 1.894406020584 = 2*cos(8*π/77) 1.835833170508 = 2*cos(10*π/77) 1.765043153573 = 2*cos(12*π/77) 1.682507065662 = 2*cos(14*π/77) = 2*cos(2*π/11) 1.588774170824 = 2*cos(16*π/77) 1.484468246009 = 2*cos(18*π/77) 1.370283429936 = 2*cos(20*π/77) 1.246979603717 = 2*cos(22*π/77) = 2*cos(2*π/7) 1.115377333974 = 2*cos(24*π/77) 0.976352412097 = 2*cos(26*π/77) 0.830830026004 = 2*cos(28*π/77) = 2*cos(4*π/11) 0.679778603165 = 2*cos(30*π/77) 0.524203365878 = 2*cos(32*π/77) 0.365139641679 = 2*cos(34*π/77) 0.203645973408 = 2*cos(36*π/77) 0.040797074783 = 2*cos(38*π/77) -0.122323321644 = 2*cos(40*π/77) -0.284629676547 = 2*cos(42*π/77) = 2*cos(6*π/11) -0.445041867913 = 2*cos(44*π/77) = 2*cos(4*π/7) -0.602492379074 = 2*cos(46*π/77) -0.755933402858 = 2*cos(48*π/77) -0.904343814565 = 2*cos(50*π/77) -1.046735967393 = 2*cos(52*π/77) -1.182162265057 = 2*cos(54*π/77) -1.309721467891 = 2*cos(56*π/77) = 2*cos(8*π/11) -1.428564690444 = 2*cos(58*π/77) -1.537901050679 = 2*cos(60*π/77) -1.637002933157 = 2*cos(62*π/77) -1.725210831202 = 2*cos(64*π/77) -1.801937735805 = 2*cos(66*π/77) = 2*cos(6*π/7) -1.866673042075 = 2*cos(68*π/77) -1.918985947229 = 2*cos(70*π/77) = 2*cos(10*π/11) -1.958528317515 = 2*cos(72*π/77) -1.985037004982 = 2*cos(74*π/77) -1.998335598689 = 2*cos(76*π/77) 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 C77 point group is 77, and the order of the principal axis (C77) is 77. The group has 39 irreducible representations. β The C77 point group is generated by one single symmetry element, C77. Therefore, it is a cyclic group. γ The lowest nonvanishing multipole moment in C77 is 2 (dipole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C77 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. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 76 cases have been combined into 38 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 38 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 38 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. η The point group is chiral, as it does not contain any mirroring operation. 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. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
C75 | ||
C76 | ||
C77 | C77v C77h D77 D77h D77d | |
C78 | ||
C79 |
This Character Table for the C77 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.