C88 E 2 C88 2 C44 2 C88^3 2 C22 2 C88^5 2 C44^3 2 C88^7 2 C11 2 C88^9 2 C44^5 2 C8 2 C22^3 2 C88^13 2 C44^7 2 C88^15 2 C11^2 2 C88^17 2 C44^9 2 C88^19 2 C22^5 2 C88^21 2 C4 2 C88^23 2 C11^3 2 C88^25 2 C44^13 2 C88^27 2 C22^7 2 C88^29 2 C44^15 2 C88^31 2 C11^4 2 C8^3 2 C44^17 2 C88^35 2 C22^9 2 C88^37 2 C44^19 2 C88^39 2 C11^5 2 C88^41 2 C44^21 2 C88^43 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 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 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1 * 2.00000 1.99490 1.97964 1.95429 1.91899 1.87390 1.81926 1.75536 1.68251 1.60108 1.51150 1.41421 1.30972 1.19856 1.08128 0.95850 0.83083 0.69893 0.56347 0.42513 0.28463 0.14268 0.00000 -0.14268 -0.28463 -0.42513 -0.56347 -0.69893 -0.83083 -0.95850 -1.08128 -1.19856 -1.30972 -1.41421 -1.51150 -1.60108 -1.68251 -1.75536 -1.81926 -1.87390 -1.91899 -1.95429 -1.97964 -1.99490 -2.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.97964 1.91899 1.81926 1.68251 1.51150 1.30972 1.08128 0.83083 0.56347 0.28463 0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 -1.97964 -1.91899 -1.81926 -1.68251 -1.51150 -1.30972 -1.08128 -0.83083 -0.56347 -0.28463 0.00000 0.28463 0.56347 0.83083 1.08128 1.30972 1.51150 1.68251 1.81926 1.91899 1.97964 2.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.95429 1.81926 1.60108 1.30972 0.95850 0.56347 0.14268 -0.28463 -0.69893 -1.08128 -1.41421 -1.68251 -1.87390 -1.97964 -1.99490 -1.91899 -1.75536 -1.51150 -1.19856 -0.83083 -0.42513 0.00000 0.42513 0.83083 1.19856 1.51150 1.75536 1.91899 1.99490 1.97964 1.87390 1.68251 1.41421 1.08128 0.69893 0.28463 -0.14268 -0.56347 -0.95850 -1.30972 -1.60108 -1.81926 -1.95429 -2.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 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 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.00000 1.87390 1.51150 0.95850 0.28463 -0.42513 -1.08128 -1.60108 -1.91899 -1.99490 -1.81926 -1.41421 -0.83083 -0.14268 0.56347 1.19856 1.68251 1.95429 1.97964 1.75536 1.30972 0.69893 0.00000 -0.69893 -1.30972 -1.75536 -1.97964 -1.95429 -1.68251 -1.19856 -0.56347 0.14268 0.83083 1.41421 1.81926 1.99490 1.91899 1.60108 1.08128 0.42513 -0.28463 -0.95850 -1.51150 -1.87390 -2.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.00000 1.81926 1.30972 0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083 0.00000 0.83083 1.51150 1.91899 1.97964 1.68251 1.08128 0.28463 -0.56347 -1.30972 -1.81926 -2.00000 -1.81926 -1.30972 -0.56347 0.28463 1.08128 1.68251 1.97964 1.91899 1.51150 0.83083 0.00000 -0.83083 -1.51150 -1.91899 -1.97964 -1.68251 -1.08128 -0.28463 0.56347 1.30972 1.81926 2.00000 ... ... ..... ....... ......... ........... TT........... E7 * 2.00000 1.75536 1.08128 0.14268 -0.83083 -1.60108 -1.97964 -1.87390 -1.30972 -0.42513 0.56347 1.41421 1.91899 1.95429 1.51150 0.69893 -0.28463 -1.19856 -1.81926 -1.99490 -1.68251 -0.95850 0.00000 0.95850 1.68251 1.99490 1.81926 1.19856 0.28463 -0.69893 -1.51150 -1.95429 -1.91899 -1.41421 -0.56347 0.42513 1.30972 1.87390 1.97964 1.60108 0.83083 -0.14268 -1.08128 -1.75536 -2.00000 ... ... ..... ....... ......... ........... ............. E8 * 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 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 1.60108 0.56347 -0.69893 -1.68251 -1.99490 -1.51150 -0.42513 0.83083 1.75536 1.97964 1.41421 0.28463 -0.95850 -1.81926 -1.95429 -1.30972 -0.14268 1.08128 1.87390 1.91899 1.19856 0.00000 -1.19856 -1.91899 -1.87390 -1.08128 0.14268 1.30972 1.95429 1.81926 0.95850 -0.28463 -1.41421 -1.97964 -1.75536 -0.83083 0.42513 1.51150 1.99490 1.68251 0.69893 -0.56347 -1.60108 -2.00000 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 1.51150 0.28463 -1.08128 -1.91899 -1.81926 -0.83083 0.56347 1.68251 1.97964 1.30972 0.00000 -1.30972 -1.97964 -1.68251 -0.56347 0.83083 1.81926 1.91899 1.08128 -0.28463 -1.51150 -2.00000 -1.51150 -0.28463 1.08128 1.91899 1.81926 0.83083 -0.56347 -1.68251 -1.97964 -1.30972 0.00000 1.30972 1.97964 1.68251 0.56347 -0.83083 -1.81926 -1.91899 -1.08128 0.28463 1.51150 2.00000 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E12 * 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 -0.83083 -1.91899 -1.68251 -0.28463 1.30972 2.00000 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 1.19856 -0.56347 -1.87390 -1.68251 -0.14268 1.51150 1.95429 0.83083 -0.95850 -1.97964 -1.41421 0.28463 1.75536 1.81926 0.42513 -1.30972 -1.99490 -1.08128 0.69893 1.91899 1.60108 0.00000 -1.60108 -1.91899 -0.69893 1.08128 1.99490 1.30972 -0.42513 -1.81926 -1.75536 -0.28463 1.41421 1.97964 0.95850 -0.83083 -1.95429 -1.51150 0.14268 1.68251 1.87390 0.56347 -1.19856 -2.00000 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 1.08128 -0.83083 -1.97964 -1.30972 0.56347 1.91899 1.51150 -0.28463 -1.81926 -1.68251 0.00000 1.68251 1.81926 0.28463 -1.51150 -1.91899 -0.56347 1.30972 1.97964 0.83083 -1.08128 -2.00000 -1.08128 0.83083 1.97964 1.30972 -0.56347 -1.91899 -1.51150 0.28463 1.81926 1.68251 0.00000 -1.68251 -1.81926 -0.28463 1.51150 1.91899 0.56347 -1.30972 -1.97964 -0.83083 1.08128 2.00000 ... ... ..... ....... ......... ........... ............. E15 * 2.00000 0.95850 -1.08128 -1.99490 -0.83083 1.19856 1.97964 0.69893 -1.30972 -1.95429 -0.56347 1.41421 1.91899 0.42513 -1.51150 -1.87390 -0.28463 1.60108 1.81926 0.14268 -1.68251 -1.75536 0.00000 1.75536 1.68251 -0.14268 -1.81926 -1.60108 0.28463 1.87390 1.51150 -0.42513 -1.91899 -1.41421 0.56347 1.95429 1.30972 -0.69893 -1.97964 -1.19856 0.83083 1.99490 1.08128 -0.95850 -2.00000 ... ... ..... ....... ......... ........... ............. E16 * 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 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 ... ... ..... ....... ......... ........... ............. E17 * 2.00000 0.69893 -1.51150 -1.75536 0.28463 1.95429 1.08128 -1.19856 -1.91899 -0.14268 1.81926 1.41421 -0.83083 -1.99490 -0.56347 1.60108 1.68251 -0.42513 -1.97964 -0.95850 1.30972 1.87390 0.00000 -1.87390 -1.30972 0.95850 1.97964 0.42513 -1.68251 -1.60108 0.56347 1.99490 0.83083 -1.41421 -1.81926 0.14268 1.91899 1.19856 -1.08128 -1.95429 -0.28463 1.75536 1.51150 -0.69893 -2.00000 ... ... ..... ....... ......... ........... ............. E18 * 2.00000 0.56347 -1.68251 -1.51150 0.83083 1.97964 0.28463 -1.81926 -1.30972 1.08128 1.91899 0.00000 -1.91899 -1.08128 1.30972 1.81926 -0.28463 -1.97964 -0.83083 1.51150 1.68251 -0.56347 -2.00000 -0.56347 1.68251 1.51150 -0.83083 -1.97964 -0.28463 1.81926 1.30972 -1.08128 -1.91899 0.00000 1.91899 1.08128 -1.30972 -1.81926 0.28463 1.97964 0.83083 -1.51150 -1.68251 0.56347 2.00000 ... ... ..... ....... ......... ........... ............. E19 * 2.00000 0.42513 -1.81926 -1.19856 1.30972 1.75536 -0.56347 -1.99490 -0.28463 1.87390 1.08128 -1.41421 -1.68251 0.69893 1.97964 0.14268 -1.91899 -0.95850 1.51150 1.60108 -0.83083 -1.95429 0.00000 1.95429 0.83083 -1.60108 -1.51150 0.95850 1.91899 -0.14268 -1.97964 -0.69893 1.68251 1.41421 -1.08128 -1.87390 0.28463 1.99490 0.56347 -1.75536 -1.30972 1.19856 1.81926 -0.42513 -2.00000 ... ... ..... ....... ......... ........... ............. E20 * 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 1.30972 1.68251 -0.83083 -1.91899 0.28463 2.00000 ... ... ..... ....... ......... ........... ............. E21 * 2.00000 0.14268 -1.97964 -0.42513 1.91899 0.69893 -1.81926 -0.95850 1.68251 1.19856 -1.51150 -1.41421 1.30972 1.60108 -1.08128 -1.75536 0.83083 1.87390 -0.56347 -1.95429 0.28463 1.99490 0.00000 -1.99490 -0.28463 1.95429 0.56347 -1.87390 -0.83083 1.75536 1.08128 -1.60108 -1.30972 1.41421 1.51150 -1.19856 -1.68251 0.95850 1.81926 -0.69893 -1.91899 0.42513 1.97964 -0.14268 -2.00000 ... ... ..... ....... ......... ........... ............. E22 * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 ... ... ..... ....... ......... ........... ............. E23 * 2.00000 -0.14268 -1.97964 0.42513 1.91899 -0.69893 -1.81926 0.95850 1.68251 -1.19856 -1.51150 1.41421 1.30972 -1.60108 -1.08128 1.75536 0.83083 -1.87390 -0.56347 1.95429 0.28463 -1.99490 0.00000 1.99490 -0.28463 -1.95429 0.56347 1.87390 -0.83083 -1.75536 1.08128 1.60108 -1.30972 -1.41421 1.51150 1.19856 -1.68251 -0.95850 1.81926 0.69893 -1.91899 -0.42513 1.97964 0.14268 -2.00000 ... ... ..... ....... ......... ........... ............. E24 * 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 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 ... ... ..... ....... ......... ........... ............. E25 * 2.00000 -0.42513 -1.81926 1.19856 1.30972 -1.75536 -0.56347 1.99490 -0.28463 -1.87390 1.08128 1.41421 -1.68251 -0.69893 1.97964 -0.14268 -1.91899 0.95850 1.51150 -1.60108 -0.83083 1.95429 0.00000 -1.95429 0.83083 1.60108 -1.51150 -0.95850 1.91899 0.14268 -1.97964 0.69893 1.68251 -1.41421 -1.08128 1.87390 0.28463 -1.99490 0.56347 1.75536 -1.30972 -1.19856 1.81926 0.42513 -2.00000 ... ... ..... ....... ......... ........... ............. E26 * 2.00000 -0.56347 -1.68251 1.51150 0.83083 -1.97964 0.28463 1.81926 -1.30972 -1.08128 1.91899 0.00000 -1.91899 1.08128 1.30972 -1.81926 -0.28463 1.97964 -0.83083 -1.51150 1.68251 0.56347 -2.00000 0.56347 1.68251 -1.51150 -0.83083 1.97964 -0.28463 -1.81926 1.30972 1.08128 -1.91899 0.00000 1.91899 -1.08128 -1.30972 1.81926 0.28463 -1.97964 0.83083 1.51150 -1.68251 -0.56347 2.00000 ... ... ..... ....... ......... ........... ............. E27 * 2.00000 -0.69893 -1.51150 1.75536 0.28463 -1.95429 1.08128 1.19856 -1.91899 0.14268 1.81926 -1.41421 -0.83083 1.99490 -0.56347 -1.60108 1.68251 0.42513 -1.97964 0.95850 1.30972 -1.87390 0.00000 1.87390 -1.30972 -0.95850 1.97964 -0.42513 -1.68251 1.60108 0.56347 -1.99490 0.83083 1.41421 -1.81926 -0.14268 1.91899 -1.19856 -1.08128 1.95429 -0.28463 -1.75536 1.51150 0.69893 -2.00000 ... ... ..... ....... ......... ........... ............. E28 * 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 -1.68251 -0.28463 1.91899 -1.30972 -0.83083 2.00000 ... ... ..... ....... ......... ........... ............. E29 * 2.00000 -0.95850 -1.08128 1.99490 -0.83083 -1.19856 1.97964 -0.69893 -1.30972 1.95429 -0.56347 -1.41421 1.91899 -0.42513 -1.51150 1.87390 -0.28463 -1.60108 1.81926 -0.14268 -1.68251 1.75536 0.00000 -1.75536 1.68251 0.14268 -1.81926 1.60108 0.28463 -1.87390 1.51150 0.42513 -1.91899 1.41421 0.56347 -1.95429 1.30972 0.69893 -1.97964 1.19856 0.83083 -1.99490 1.08128 0.95850 -2.00000 ... ... ..... ....... ......... ........... ............. E30 * 2.00000 -1.08128 -0.83083 1.97964 -1.30972 -0.56347 1.91899 -1.51150 -0.28463 1.81926 -1.68251 0.00000 1.68251 -1.81926 0.28463 1.51150 -1.91899 0.56347 1.30972 -1.97964 0.83083 1.08128 -2.00000 1.08128 0.83083 -1.97964 1.30972 0.56347 -1.91899 1.51150 0.28463 -1.81926 1.68251 0.00000 -1.68251 1.81926 -0.28463 -1.51150 1.91899 -0.56347 -1.30972 1.97964 -0.83083 -1.08128 2.00000 ... ... ..... ....... ......... ........... ............. E31 * 2.00000 -1.19856 -0.56347 1.87390 -1.68251 0.14268 1.51150 -1.95429 0.83083 0.95850 -1.97964 1.41421 0.28463 -1.75536 1.81926 -0.42513 -1.30972 1.99490 -1.08128 -0.69893 1.91899 -1.60108 0.00000 1.60108 -1.91899 0.69893 1.08128 -1.99490 1.30972 0.42513 -1.81926 1.75536 -0.28463 -1.41421 1.97964 -0.95850 -0.83083 1.95429 -1.51150 -0.14268 1.68251 -1.87390 0.56347 1.19856 -2.00000 ... ... ..... ....... ......... ........... ............. E32 * 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 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 ... ... ..... ....... ......... ........... ............. E33 * 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 ... ... ..... ....... ......... ........... ............. E34 * 2.00000 -1.51150 0.28463 1.08128 -1.91899 1.81926 -0.83083 -0.56347 1.68251 -1.97964 1.30972 0.00000 -1.30972 1.97964 -1.68251 0.56347 0.83083 -1.81926 1.91899 -1.08128 -0.28463 1.51150 -2.00000 1.51150 -0.28463 -1.08128 1.91899 -1.81926 0.83083 0.56347 -1.68251 1.97964 -1.30972 0.00000 1.30972 -1.97964 1.68251 -0.56347 -0.83083 1.81926 -1.91899 1.08128 0.28463 -1.51150 2.00000 ... ... ..... ....... ......... ........... ............. E35 * 2.00000 -1.60108 0.56347 0.69893 -1.68251 1.99490 -1.51150 0.42513 0.83083 -1.75536 1.97964 -1.41421 0.28463 0.95850 -1.81926 1.95429 -1.30972 0.14268 1.08128 -1.87390 1.91899 -1.19856 0.00000 1.19856 -1.91899 1.87390 -1.08128 -0.14268 1.30972 -1.95429 1.81926 -0.95850 -0.28463 1.41421 -1.97964 1.75536 -0.83083 -0.42513 1.51150 -1.99490 1.68251 -0.69893 -0.56347 1.60108 -2.00000 ... ... ..... ....... ......... ........... ............. E36 * 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 1.91899 -1.30972 0.28463 0.83083 -1.68251 2.00000 ... ... ..... ....... ......... ........... ............. E37 * 2.00000 -1.75536 1.08128 -0.14268 -0.83083 1.60108 -1.97964 1.87390 -1.30972 0.42513 0.56347 -1.41421 1.91899 -1.95429 1.51150 -0.69893 -0.28463 1.19856 -1.81926 1.99490 -1.68251 0.95850 0.00000 -0.95850 1.68251 -1.99490 1.81926 -1.19856 0.28463 0.69893 -1.51150 1.95429 -1.91899 1.41421 -0.56347 -0.42513 1.30972 -1.87390 1.97964 -1.60108 0.83083 0.14268 -1.08128 1.75536 -2.00000 ... ... ..... ....... ......... ........... ............. E38 * 2.00000 -1.81926 1.30972 -0.56347 -0.28463 1.08128 -1.68251 1.97964 -1.91899 1.51150 -0.83083 0.00000 0.83083 -1.51150 1.91899 -1.97964 1.68251 -1.08128 0.28463 0.56347 -1.30972 1.81926 -2.00000 1.81926 -1.30972 0.56347 0.28463 -1.08128 1.68251 -1.97964 1.91899 -1.51150 0.83083 0.00000 -0.83083 1.51150 -1.91899 1.97964 -1.68251 1.08128 -0.28463 -0.56347 1.30972 -1.81926 2.00000 ... ... ..... ....... ......... ........... ............. E39 * 2.00000 -1.87390 1.51150 -0.95850 0.28463 0.42513 -1.08128 1.60108 -1.91899 1.99490 -1.81926 1.41421 -0.83083 0.14268 0.56347 -1.19856 1.68251 -1.95429 1.97964 -1.75536 1.30972 -0.69893 0.00000 0.69893 -1.30972 1.75536 -1.97964 1.95429 -1.68251 1.19856 -0.56347 -0.14268 0.83083 -1.41421 1.81926 -1.99490 1.91899 -1.60108 1.08128 -0.42513 -0.28463 0.95850 -1.51150 1.87390 -2.00000 ... ... ..... ....... ......... ........... ............. E40 * 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 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 ... ... ..... ....... ......... ........... ............. E41 * 2.00000 -1.95429 1.81926 -1.60108 1.30972 -0.95850 0.56347 -0.14268 -0.28463 0.69893 -1.08128 1.41421 -1.68251 1.87390 -1.97964 1.99490 -1.91899 1.75536 -1.51150 1.19856 -0.83083 0.42513 0.00000 -0.42513 0.83083 -1.19856 1.51150 -1.75536 1.91899 -1.99490 1.97964 -1.87390 1.68251 -1.41421 1.08128 -0.69893 0.28463 0.14268 -0.56347 0.95850 -1.30972 1.60108 -1.81926 1.95429 -2.00000 ... ... ..... ....... ......... ........... ............. E42 * 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 1.97964 -1.91899 1.81926 -1.68251 1.51150 -1.30972 1.08128 -0.83083 0.56347 -0.28463 0.00000 0.28463 -0.56347 0.83083 -1.08128 1.30972 -1.51150 1.68251 -1.81926 1.91899 -1.97964 2.00000 ... ... ..... ....... ......... ........... ............. E43 * 2.00000 -1.99490 1.97964 -1.95429 1.91899 -1.87390 1.81926 -1.75536 1.68251 -1.60108 1.51150 -1.41421 1.30972 -1.19856 1.08128 -0.95850 0.83083 -0.69893 0.56347 -0.42513 0.28463 -0.14268 0.00000 0.14268 -0.28463 0.42513 -0.56347 0.69893 -0.83083 0.95850 -1.08128 1.19856 -1.30972 1.41421 -1.51150 1.60108 -1.68251 1.75536 -1.81926 1.87390 -1.91899 1.95429 -1.97964 1.99490 -2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.994904229221 = 2*cos(2*π/88) = 2*cos(π/44) 1.979642883762 = 2*cos(4*π/88) = 2*cos(π/22) 1.954293731942 = 2*cos(6*π/88) = 2*cos(3*π/44) 1.918985947229 = 2*cos(8*π/88) = 2*cos(π/11) 1.873899450000 = 2*cos(10*π/88) = 2*cos(5*π/44) 1.819263990709 = 2*cos(12*π/88) = 2*cos(3*π/22) 1.755357979135 = 2*cos(14*π/88) = 2*cos(7*π/44) 1.682507065662 = 2*cos(16*π/88) = 2*cos(2*π/11) 1.601082481849 = 2*cos(18*π/88) = 2*cos(9*π/44) 1.511499148709 = 2*cos(20*π/88) = 2*cos(5*π/22) 1.414213562373 = 2*cos(22*π/88) = 2*cos(π/4) = √2 1.309721467891 = 2*cos(24*π/88) = 2*cos(3*π/11) 1.198555333023 = 2*cos(26*π/88) = 2*cos(13*π/44) 1.081281634911 = 2*cos(28*π/88) = 2*cos(7*π/22) 0.958497973440 = 2*cos(30*π/88) = 2*cos(15*π/44) 0.830830026004 = 2*cos(32*π/88) = 2*cos(4*π/11) 0.698928359198 = 2*cos(34*π/88) = 2*cos(17*π/44) 0.563465113683 = 2*cos(36*π/88) = 2*cos(9*π/22) 0.425130579106 = 2*cos(38*π/88) = 2*cos(19*π/44) 0.284629676547 = 2*cos(40*π/88) = 2*cos(5*π/11) 0.142678366398 = 2*cos(42*π/88) = 2*cos(21*π/44) 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 C88 point group is 88, and the order of the principal axis (C88) is 88. The group has 45 irreducible representations. β The C88 point group is isomorphic to S88. γ The C88 point group is generated by one single symmetry element, C88. Therefore, it is a cyclic group. δ The lowest nonvanishing multipole moment in C88 is 2 (dipole moment). ε This is an Abelian point group (the commutative law holds between all symmetry operations). The C88 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 86 cases have been combined into 43 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 43 pairs of left and right rotations have been combined into one two-membered pseudo-class each. η The 43 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, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
| C86 | ||
| C87 | ||
| C88 | C88v C88h D88 D88h D88d S88 | |
| C89 | ||
| C90 |
This Character Table for the C88 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.