D81 E 2 C81 2 C81^2 2 C27 2 C81^4 2 C81^5 2 C27^2 2 C81^7 2 C81^8 2 C9 2 C81^10 2 C81^11 2 C27^4 2 C81^13 2 C81^14 2 C27^5 2 C81^16 2 C81^17 2 C9^2 2 C81^19 2 C81^20 2 C27^7 2 C81^22 2 C81^23 2 C27^8 2 C81^25 2 C81^26 2 C3 2 C81^28 2 C81^29 2 C27^10 2 C81^31 2 C81^32 2 C27^11 2 C81^34 2 C81^35 2 C9^4 2 C81^37 2 C81^38 2 C27^13 2 C81^40 81 C2' <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 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 A2 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 ............. E1 2.00000 1.99399 1.97598 1.94609 1.90450 1.85145 1.78727 1.71233 1.62710 1.53209 1.42786 1.31504 1.19432 1.06641 0.93209 0.79216 0.64747 0.49888 0.34730 0.19362 0.03878 -0.11629 -0.27066 -0.42341 -0.57361 -0.72036 -0.86277 -1.00000 -1.13121 -1.25562 -1.37248 -1.48109 -1.58079 -1.67098 -1.75112 -1.82073 -1.87939 -1.92674 -1.96251 -1.98648 -1.99850 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.97598 1.90450 1.78727 1.62710 1.42786 1.19432 0.93209 0.64747 0.34730 0.03878 -0.27066 -0.57361 -0.86277 -1.13121 -1.37248 -1.58079 -1.75112 -1.87939 -1.96251 -1.99850 -1.98648 -1.92674 -1.82073 -1.67098 -1.48109 -1.25562 -1.00000 -0.72036 -0.42341 -0.11629 0.19362 0.49888 0.79216 1.06641 1.31504 1.53209 1.71233 1.85145 1.94609 1.99399 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629 0.34730 0.79216 1.19432 1.53209 1.78727 1.94609 2.00000 1.94609 1.78727 1.53209 1.19432 0.79216 0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.90450 1.62710 1.19432 0.64747 0.03878 -0.57361 -1.13121 -1.58079 -1.87939 -1.99850 -1.92674 -1.67098 -1.25562 -0.72036 -0.11629 0.49888 1.06641 1.53209 1.85145 1.99399 1.94609 1.71233 1.31504 0.79216 0.19362 -0.42341 -1.00000 -1.48109 -1.82073 -1.98648 -1.96251 -1.75112 -1.37248 -0.86277 -0.27066 0.34730 0.93209 1.42786 1.78727 1.97598 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.85145 1.42786 0.79216 0.03878 -0.72036 -1.37248 -1.82073 -1.99850 -1.87939 -1.48109 -0.86277 -0.11629 0.64747 1.31504 1.78727 1.99399 1.90450 1.53209 0.93209 0.19362 -0.57361 -1.25562 -1.75112 -1.98648 -1.92674 -1.58079 -1.00000 -0.27066 0.49888 1.19432 1.71233 1.97598 1.94609 1.62710 1.06641 0.34730 -0.42341 -1.13121 -1.67098 -1.96251 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.78727 1.19432 0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629 0.79216 1.53209 1.94609 1.94609 1.53209 0.79216 -0.11629 -1.00000 -1.67098 -1.98648 -1.87939 -1.37248 -0.57361 0.34730 1.19432 1.78727 2.00000 1.78727 1.19432 0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629 0.79216 1.53209 1.94609 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.71233 0.93209 -0.11629 -1.13121 -1.82073 -1.98648 -1.58079 -0.72036 0.34730 1.31504 1.90450 1.94609 1.42786 0.49888 -0.57361 -1.48109 -1.96251 -1.87939 -1.25562 -0.27066 0.79216 1.62710 1.99399 1.78727 1.06641 0.03878 -1.00000 -1.75112 -1.99850 -1.67098 -0.86277 0.19362 1.19432 1.85145 1.97598 1.53209 0.64747 -0.42341 -1.37248 -1.92674 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.62710 0.64747 -0.57361 -1.58079 -1.99850 -1.67098 -0.72036 0.49888 1.53209 1.99399 1.71233 0.79216 -0.42341 -1.48109 -1.98648 -1.75112 -0.86277 0.34730 1.42786 1.97598 1.78727 0.93209 -0.27066 -1.37248 -1.96251 -1.82073 -1.00000 0.19362 1.31504 1.94609 1.85145 1.06641 -0.11629 -1.25562 -1.92674 -1.87939 -1.13121 0.03878 1.19432 1.90450 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 -1.87939 -1.00000 0.34730 1.53209 2.00000 1.53209 0.34730 -1.00000 -1.87939 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.42786 0.03878 -1.37248 -1.99850 -1.48109 -0.11629 1.31504 1.99399 1.53209 0.19362 -1.25562 -1.98648 -1.58079 -0.27066 1.19432 1.97598 1.62710 0.34730 -1.13121 -1.96251 -1.67098 -0.42341 1.06641 1.94609 1.71233 0.49888 -1.00000 -1.92674 -1.75112 -0.57361 0.93209 1.90450 1.78727 0.64747 -0.86277 -1.87939 -1.82073 -0.72036 0.79216 1.85145 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.31504 -0.27066 -1.67098 -1.92674 -0.86277 0.79216 1.90450 1.71233 0.34730 -1.25562 -1.99850 -1.37248 0.19362 1.62710 1.94609 0.93209 -0.72036 -1.87939 -1.75112 -0.42341 1.19432 1.99399 1.42786 -0.11629 -1.58079 -1.96251 -1.00000 0.64747 1.85145 1.78727 0.49888 -1.13121 -1.98648 -1.48109 0.03878 1.53209 1.97598 1.06641 -0.57361 -1.82073 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 1.19432 -0.57361 -1.87939 -1.67098 -0.11629 1.53209 1.94609 0.79216 -1.00000 -1.98648 -1.37248 0.34730 1.78727 1.78727 0.34730 -1.37248 -1.98648 -1.00000 0.79216 1.94609 1.53209 -0.11629 -1.67098 -1.87939 -0.57361 1.19432 2.00000 1.19432 -0.57361 -1.87939 -1.67098 -0.11629 1.53209 1.94609 0.79216 -1.00000 -1.98648 -1.37248 0.34730 1.78727 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 1.06641 -0.86277 -1.98648 -1.25562 0.64747 1.94609 1.42786 -0.42341 -1.87939 -1.58079 0.19362 1.78727 1.71233 0.03878 -1.67098 -1.82073 -0.27066 1.53209 1.90450 0.49888 -1.37248 -1.96251 -0.72036 1.19432 1.99399 0.93209 -1.00000 -1.99850 -1.13121 0.79216 1.97598 1.31504 -0.57361 -1.92674 -1.48109 0.34730 1.85145 1.62710 -0.11629 -1.75112 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.93209 -1.13121 -1.98648 -0.72036 1.31504 1.94609 0.49888 -1.48109 -1.87939 -0.27066 1.62710 1.78727 0.03878 -1.75112 -1.67098 0.19362 1.85145 1.53209 -0.42341 -1.92674 -1.37248 0.64747 1.97598 1.19432 -0.86277 -1.99850 -1.00000 1.06641 1.99399 0.79216 -1.25562 -1.96251 -0.57361 1.42786 1.90450 0.34730 -1.58079 -1.82073 -0.11629 1.71233 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.79216 -1.37248 -1.87939 -0.11629 1.78727 1.53209 -0.57361 -1.98648 -1.00000 1.19432 1.94609 0.34730 -1.67098 -1.67098 0.34730 1.94609 1.19432 -1.00000 -1.98648 -0.57361 1.53209 1.78727 -0.11629 -1.87939 -1.37248 0.79216 2.00000 0.79216 -1.37248 -1.87939 -0.11629 1.78727 1.53209 -0.57361 -1.98648 -1.00000 1.19432 1.94609 0.34730 -1.67098 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.64747 -1.58079 -1.67098 0.49888 1.99399 0.79216 -1.48109 -1.75112 0.34730 1.97598 0.93209 -1.37248 -1.82073 0.19362 1.94609 1.06641 -1.25562 -1.87939 0.03878 1.90450 1.19432 -1.13121 -1.92674 -0.11629 1.85145 1.31504 -1.00000 -1.96251 -0.27066 1.78727 1.42786 -0.86277 -1.98648 -0.42341 1.71233 1.53209 -0.72036 -1.99850 -0.57361 1.62710 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.49888 -1.75112 -1.37248 1.06641 1.90450 -0.11629 -1.96251 -0.86277 1.53209 1.62710 -0.72036 -1.98648 -0.27066 1.85145 1.19432 -1.25562 -1.82073 0.34730 1.99399 0.64747 -1.67098 -1.48109 0.93209 1.94609 0.03878 -1.92674 -1.00000 1.42786 1.71233 -0.57361 -1.99850 -0.42341 1.78727 1.31504 -1.13121 -1.87939 0.19362 1.97598 0.79216 -1.58079 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 1.53209 -1.00000 -1.87939 0.34730 2.00000 0.34730 -1.87939 -1.00000 1.53209 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 0.19362 -1.96251 -0.57361 1.85145 0.93209 -1.67098 -1.25562 1.42786 1.53209 -1.13121 -1.75112 0.79216 1.90450 -0.42341 -1.98648 0.03878 1.99399 0.34730 -1.92674 -0.72036 1.78727 1.06641 -1.58079 -1.37248 1.31504 1.62710 -1.00000 -1.82073 0.64747 1.94609 -0.27066 -1.99850 -0.11629 1.97598 0.49888 -1.87939 -0.86277 1.71233 1.19432 -1.48109 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 0.03878 -1.99850 -0.11629 1.99399 0.19362 -1.98648 -0.27066 1.97598 0.34730 -1.96251 -0.42341 1.94609 0.49888 -1.92674 -0.57361 1.90450 0.64747 -1.87939 -0.72036 1.85145 0.79216 -1.82073 -0.86277 1.78727 0.93209 -1.75112 -1.00000 1.71233 1.06641 -1.67098 -1.13121 1.62710 1.19432 -1.58079 -1.25562 1.53209 1.31504 -1.48109 -1.37248 1.42786 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.11629 -1.98648 0.34730 1.94609 -0.57361 -1.87939 0.79216 1.78727 -1.00000 -1.67098 1.19432 1.53209 -1.37248 -1.37248 1.53209 1.19432 -1.67098 -1.00000 1.78727 0.79216 -1.87939 -0.57361 1.94609 0.34730 -1.98648 -0.11629 2.00000 -0.11629 -1.98648 0.34730 1.94609 -0.57361 -1.87939 0.79216 1.78727 -1.00000 -1.67098 1.19432 1.53209 -1.37248 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.27066 -1.92674 0.79216 1.71233 -1.25562 -1.37248 1.62710 0.93209 -1.87939 -0.42341 1.99399 -0.11629 -1.96251 0.64747 1.78727 -1.13121 -1.48109 1.53209 1.06641 -1.82073 -0.57361 1.97598 0.03878 -1.98648 0.49888 1.85145 -1.00000 -1.58079 1.42786 1.19432 -1.75112 -0.72036 1.94609 0.19362 -1.99850 0.34730 1.90450 -0.86277 -1.67098 1.31504 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -0.42341 -1.82073 1.19432 1.31504 -1.75112 -0.57361 1.99399 -0.27066 -1.87939 1.06641 1.42786 -1.67098 -0.72036 1.97598 -0.11629 -1.92674 0.93209 1.53209 -1.58079 -0.86277 1.94609 0.03878 -1.96251 0.79216 1.62710 -1.48109 -1.00000 1.90450 0.19362 -1.98648 0.64747 1.71233 -1.37248 -1.13121 1.85145 0.34730 -1.99850 0.49888 1.78727 -1.25562 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -0.57361 -1.67098 1.53209 0.79216 -1.98648 0.34730 1.78727 -1.37248 -1.00000 1.94609 -0.11629 -1.87939 1.19432 1.19432 -1.87939 -0.11629 1.94609 -1.00000 -1.37248 1.78727 0.34730 -1.98648 0.79216 1.53209 -1.67098 -0.57361 2.00000 -0.57361 -1.67098 1.53209 0.79216 -1.98648 0.34730 1.78727 -1.37248 -1.00000 1.94609 -0.11629 -1.87939 1.19432 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -0.72036 -1.48109 1.78727 0.19362 -1.92674 1.19432 1.06641 -1.96251 0.34730 1.71233 -1.58079 -0.57361 1.99399 -0.86277 -1.37248 1.85145 0.03878 -1.87939 1.31504 0.93209 -1.98648 0.49888 1.62710 -1.67098 -0.42341 1.97598 -1.00000 -1.25562 1.90450 -0.11629 -1.82073 1.42786 0.79216 -1.99850 0.64747 1.53209 -1.75112 -0.27066 1.94609 -1.13121 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -0.86277 -1.25562 1.94609 -0.42341 -1.58079 1.78727 0.03878 -1.82073 1.53209 0.49888 -1.96251 1.19432 0.93209 -1.99850 0.79216 1.31504 -1.92674 0.34730 1.62710 -1.75112 -0.11629 1.85145 -1.48109 -0.57361 1.97598 -1.13121 -1.00000 1.99399 -0.72036 -1.37248 1.90450 -0.27066 -1.67098 1.71233 0.19362 -1.87939 1.42786 0.64747 -1.98648 1.06641 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.13121 -0.72036 1.94609 -1.48109 -0.27066 1.78727 -1.75112 0.19362 1.53209 -1.92674 0.64747 1.19432 -1.99850 1.06641 0.79216 -1.96251 1.42786 0.34730 -1.82073 1.71233 -0.11629 -1.58079 1.90450 -0.57361 -1.25562 1.99399 -1.00000 -0.86277 1.97598 -1.37248 -0.42341 1.85145 -1.67098 0.03878 1.62710 -1.87939 0.49888 1.31504 -1.98648 0.93209 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.25562 -0.42341 1.78727 -1.82073 0.49888 1.19432 -1.99850 1.31504 0.34730 -1.75112 1.85145 -0.57361 -1.13121 1.99399 -1.37248 -0.27066 1.71233 -1.87939 0.64747 1.06641 -1.98648 1.42786 0.19362 -1.67098 1.90450 -0.72036 -1.00000 1.97598 -1.48109 -0.11629 1.62710 -1.92674 0.79216 0.93209 -1.96251 1.53209 0.03878 -1.58079 1.94609 -0.86277 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.37248 -0.11629 1.53209 -1.98648 1.19432 0.34730 -1.67098 1.94609 -1.00000 -0.57361 1.78727 -1.87939 0.79216 0.79216 -1.87939 1.78727 -0.57361 -1.00000 1.94609 -1.67098 0.34730 1.19432 -1.98648 1.53209 -0.11629 -1.37248 2.00000 -1.37248 -0.11629 1.53209 -1.98648 1.19432 0.34730 -1.67098 1.94609 -1.00000 -0.57361 1.78727 -1.87939 0.79216 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.48109 0.19362 1.19432 -1.96251 1.71233 -0.57361 -0.86277 1.85145 -1.87939 0.93209 0.49888 -1.67098 1.97598 -1.25562 -0.11629 1.42786 -1.99850 1.53209 -0.27066 -1.13121 1.94609 -1.75112 0.64747 0.79216 -1.82073 1.90450 -1.00000 -0.42341 1.62710 -1.98648 1.31504 0.03878 -1.37248 1.99399 -1.58079 0.34730 1.06641 -1.92674 1.78727 -0.72036 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.58079 0.49888 0.79216 -1.75112 1.97598 -1.37248 0.19362 1.06641 -1.87939 1.90450 -1.13121 -0.11629 1.31504 -1.96251 1.78727 -0.86277 -0.42341 1.53209 -1.99850 1.62710 -0.57361 -0.72036 1.71233 -1.98648 1.42786 -0.27066 -1.00000 1.85145 -1.92674 1.19432 0.03878 -1.25562 1.94609 -1.82073 0.93209 0.34730 -1.48109 1.99399 -1.67098 0.64747 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.67098 0.79216 0.34730 -1.37248 1.94609 -1.87939 1.19432 -0.11629 -1.00000 1.78727 -1.98648 1.53209 -0.57361 -0.57361 1.53209 -1.98648 1.78727 -1.00000 -0.11629 1.19432 -1.87939 1.94609 -1.37248 0.34730 0.79216 -1.67098 2.00000 -1.67098 0.79216 0.34730 -1.37248 1.94609 -1.87939 1.19432 -0.11629 -1.00000 1.78727 -1.98648 1.53209 -0.57361 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.75112 1.06641 -0.11629 -0.86277 1.62710 -1.98648 1.85145 -1.25562 0.34730 0.64747 -1.48109 1.94609 -1.92674 1.42786 -0.57361 -0.42341 1.31504 -1.87939 1.97598 -1.58079 0.79216 0.19362 -1.13121 1.78727 -1.99850 1.71233 -1.00000 0.03878 0.93209 -1.67098 1.99399 -1.82073 1.19432 -0.27066 -0.72036 1.53209 -1.96251 1.90450 -1.37248 0.49888 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.82073 1.31504 -0.57361 -0.27066 1.06641 -1.67098 1.97598 -1.92674 1.53209 -0.86277 0.03878 0.79216 -1.48109 1.90450 -1.98648 1.71233 -1.13121 0.34730 0.49888 -1.25562 1.78727 -1.99850 1.85145 -1.37248 0.64747 0.19362 -1.00000 1.62710 -1.96251 1.94609 -1.58079 0.93209 -0.11629 -0.72036 1.42786 -1.87939 1.99399 -1.75112 1.19432 -0.42341 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.34730 -1.00000 1.53209 -1.87939 2.00000 -1.87939 1.53209 -1.00000 0.34730 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.92674 1.71233 -1.37248 0.93209 -0.42341 -0.11629 0.64747 -1.13121 1.53209 -1.82073 1.97598 -1.98648 1.85145 -1.58079 1.19432 -0.72036 0.19362 0.34730 -0.86277 1.31504 -1.67098 1.90450 -1.99850 1.94609 -1.75112 1.42786 -1.00000 0.49888 0.03878 -0.57361 1.06641 -1.48109 1.78727 -1.96251 1.99399 -1.87939 1.62710 -1.25562 0.79216 -0.27066 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.96251 1.85145 -1.67098 1.42786 -1.13121 0.79216 -0.42341 0.03878 0.34730 -0.72036 1.06641 -1.37248 1.62710 -1.82073 1.94609 -1.99850 1.97598 -1.87939 1.71233 -1.48109 1.19432 -0.86277 0.49888 -0.11629 -0.27066 0.64747 -1.00000 1.31504 -1.58079 1.78727 -1.92674 1.99399 -1.98648 1.90450 -1.75112 1.53209 -1.25562 0.93209 -0.57361 0.19362 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.98648 1.94609 -1.87939 1.78727 -1.67098 1.53209 -1.37248 1.19432 -1.00000 0.79216 -0.57361 0.34730 -0.11629 -0.11629 0.34730 -0.57361 0.79216 -1.00000 1.19432 -1.37248 1.53209 -1.67098 1.78727 -1.87939 1.94609 -1.98648 2.00000 -1.98648 1.94609 -1.87939 1.78727 -1.67098 1.53209 -1.37248 1.19432 -1.00000 0.79216 -0.57361 0.34730 -0.11629 0.00000 ... ... ..... ....... ......... ........... ............. E40 2.00000 -1.99850 1.99399 -1.98648 1.97598 -1.96251 1.94609 -1.92674 1.90450 -1.87939 1.85145 -1.82073 1.78727 -1.75112 1.71233 -1.67098 1.62710 -1.58079 1.53209 -1.48109 1.42786 -1.37248 1.31504 -1.25562 1.19432 -1.13121 1.06641 -1.00000 0.93209 -0.86277 0.79216 -0.72036 0.64747 -0.57361 0.49888 -0.42341 0.34730 -0.27066 0.19362 -0.11629 0.03878 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.993985882336 = 2*cos(2*π/81) 1.975979698954 = 2*cos(4*π/81) 1.946089741160 = 2*cos(6*π/81) = 2*cos(2*π/27) 1.904495770677 = 2*cos(8*π/81) 1.851447938538 = 2*cos(10*π/81) 1.787265280647 = 2*cos(12*π/81) = 2*cos(4*π/27) 1.712333799061 = 2*cos(14*π/81) 1.627104140526 = 2*cos(16*π/81) 1.532088886238 = 2*cos(18*π/81) = 2*cos(2*π/9) 1.427859469116 = 2*cos(20*π/81) 1.315042737138 = 2*cos(22*π/81) 1.194317183406 = 2*cos(24*π/81) = 2*cos(8*π/27) 1.066408865603 = 2*cos(26*π/81) 0.932087039405 = 2*cos(28*π/81) 0.792159532078 = 2*cos(30*π/81) = 2*cos(10*π/27) 0.647467884117 = 2*cos(32*π/81) 0.498882288116 = 2*cos(34*π/81) 0.347296355334 = 2*cos(36*π/81) = 2*cos(4*π/9) 0.193621741406 = 2*cos(38*π/81) 0.038782663544 = 2*cos(40*π/81) -0.116289657821 = 2*cos(42*π/81) = 2*cos(14*π/27) -0.270662599500 = 2*cos(44*π/81) -0.423407744459 = 2*cos(46*π/81) -0.573606465422 = 2*cos(48*π/81) = 2*cos(16*π/27) -0.720355449609 = 2*cos(50*π/81) -0.862772131363 = 2*cos(52*π/81) -1.131213750973 = 2*cos(56*π/81) -1.255624249344 = 2*cos(58*π/81) -1.372483275737 = 2*cos(60*π/81) = 2*cos(20*π/27) -1.481088026218 = 2*cos(62*π/81) -1.580785339038 = 2*cos(64*π/81) -1.670975622826 = 2*cos(66*π/81) = 2*cos(22*π/27) -1.751116462604 = 2*cos(68*π/81) -1.820725881932 = 2*cos(70*π/81) -1.879385241572 = 2*cos(72*π/81) = 2*cos(8*π/9) -1.926741757232 = 2*cos(74*π/81) -1.962510621255 = 2*cos(76*π/81) -1.986476715484 = 2*cos(78*π/81) = 2*cos(26*π/27) -1.998495905008 = 2*cos(80*π/81) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 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 D81 point group is 162, and the order of the principal axis (C81) is 81. The group has 42 irreducible representations. β The D81 point group is isomorphic to C81v. γ The D81 point group is generated by two symmetry elements, C81 and a perpendicular C2′. Also, the group may be generated from two C2′ axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. ε The lowest nonvanishing multipole moment in D81 is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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. 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. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
D79 | ||
D80 | ||
C81 C81v C81h | D81 | D81h D81d |
D82 | ||
D83 |
This Character Table for the D81 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.