C128 E 2 C128 2 C64 2 C128^3 2 C32 2 C128^5 2 C64^3 2 C128^7 2 C16 2 C128^9 2 C64^5 2 C128^11 2 C32^3 2 C128^13 2 C64^7 2 C128^15 2 C8 2 C128^17 2 C64^9 2 C128^19 2 C32^5 2 C128^21 2 C64^11 2 C128^23 2 C16^3 2 C128^25 2 C64^13 2 C128^27 2 C32^7 2 C128^29 2 C64^15 2 C128^31 2 C4 2 C128^33 2 C64^17 2 C128^35 2 C32^9 2 C128^37 2 C64^19 2 C128^39 2 C16^5 2 C128^41 2 C64^21 2 C128^43 2 C32^11 2 C128^45 2 C64^23 2 C128^47 2 C8^3 2 C128^49 2 C64^25 2 C128^51 2 C32^13 2 C128^53 2 C64^27 2 C128^55 2 C16^7 2 C128^57 2 C64^29 2 C128^59 2 C32^15 2 C128^61 2 C64^31 2 C128^63 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 1.99759 1.99037 1.97835 1.96157 1.94006 1.91388 1.88309 1.84776 1.80798 1.76384 1.71546 1.66294 1.60642 1.54602 1.48190 1.41421 1.34312 1.26879 1.19140 1.11114 1.02821 0.94279 0.85511 0.76537 0.67378 0.58057 0.48596 0.39018 0.29346 0.19603 0.09814 0.00000 -0.09814 -0.19603 -0.29346 -0.39018 -0.48596 -0.58057 -0.67378 -0.76537 -0.85511 -0.94279 -1.02821 -1.11114 -1.19140 -1.26879 -1.34312 -1.41421 -1.48190 -1.54602 -1.60642 -1.66294 -1.71546 -1.76384 -1.80798 -1.84776 -1.88309 -1.91388 -1.94006 -1.96157 -1.97835 -1.99037 -1.99759 -2.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.99037 1.96157 1.91388 1.84776 1.76384 1.66294 1.54602 1.41421 1.26879 1.11114 0.94279 0.76537 0.58057 0.39018 0.19603 0.00000 -0.19603 -0.39018 -0.58057 -0.76537 -0.94279 -1.11114 -1.26879 -1.41421 -1.54602 -1.66294 -1.76384 -1.84776 -1.91388 -1.96157 -1.99037 -2.00000 -1.99037 -1.96157 -1.91388 -1.84776 -1.76384 -1.66294 -1.54602 -1.41421 -1.26879 -1.11114 -0.94279 -0.76537 -0.58057 -0.39018 -0.19603 0.00000 0.19603 0.39018 0.58057 0.76537 0.94279 1.11114 1.26879 1.41421 1.54602 1.66294 1.76384 1.84776 1.91388 1.96157 1.99037 2.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.97835 1.91388 1.80798 1.66294 1.48190 1.26879 1.02821 0.76537 0.48596 0.19603 -0.09814 -0.39018 -0.67378 -0.94279 -1.19140 -1.41421 -1.60642 -1.76384 -1.88309 -1.96157 -1.99759 -1.99037 -1.94006 -1.84776 -1.71546 -1.54602 -1.34312 -1.11114 -0.85511 -0.58057 -0.29346 0.00000 0.29346 0.58057 0.85511 1.11114 1.34312 1.54602 1.71546 1.84776 1.94006 1.99037 1.99759 1.96157 1.88309 1.76384 1.60642 1.41421 1.19140 0.94279 0.67378 0.39018 0.09814 -0.19603 -0.48596 -0.76537 -1.02821 -1.26879 -1.48190 -1.66294 -1.80798 -1.91388 -1.97835 -2.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... 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E9 * 2.00000 1.80798 1.26879 0.48596 -0.39018 -1.19140 -1.76384 -1.99759 -1.84776 -1.34312 -0.58057 0.29346 1.11114 1.71546 1.99037 1.88309 1.41421 0.67378 -0.19603 -1.02821 -1.66294 -1.97835 -1.91388 -1.48190 -0.76537 0.09814 0.94279 1.60642 1.96157 1.94006 1.54602 0.85511 0.00000 -0.85511 -1.54602 -1.94006 -1.96157 -1.60642 -0.94279 -0.09814 0.76537 1.48190 1.91388 1.97835 1.66294 1.02821 0.19603 -0.67378 -1.41421 -1.88309 -1.99037 -1.71546 -1.11114 -0.29346 0.58057 1.34312 1.84776 1.99759 1.76384 1.19140 0.39018 -0.48596 -1.26879 -1.80798 -2.00000 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 1.76384 1.11114 0.19603 -0.76537 -1.54602 -1.96157 -1.91388 -1.41421 -0.58057 0.39018 1.26879 1.84776 1.99037 1.66294 0.94279 0.00000 -0.94279 -1.66294 -1.99037 -1.84776 -1.26879 -0.39018 0.58057 1.41421 1.91388 1.96157 1.54602 0.76537 -0.19603 -1.11114 -1.76384 -2.00000 -1.76384 -1.11114 -0.19603 0.76537 1.54602 1.96157 1.91388 1.41421 0.58057 -0.39018 -1.26879 -1.84776 -1.99037 -1.66294 -0.94279 0.00000 0.94279 1.66294 1.99037 1.84776 1.26879 0.39018 -0.58057 -1.41421 -1.91388 -1.96157 -1.54602 -0.76537 0.19603 1.11114 1.76384 2.00000 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 1.71546 0.94279 -0.09814 -1.11114 -1.80798 -1.99037 -1.60642 -0.76537 0.29346 1.26879 1.88309 1.96157 1.48190 0.58057 -0.48596 -1.41421 -1.94006 -1.91388 -1.34312 -0.39018 0.67378 1.54602 1.97835 1.84776 1.19140 0.19603 -0.85511 -1.66294 -1.99759 -1.76384 -1.02821 0.00000 1.02821 1.76384 1.99759 1.66294 0.85511 -0.19603 -1.19140 -1.84776 -1.97835 -1.54602 -0.67378 0.39018 1.34312 1.91388 1.94006 1.41421 0.48596 -0.58057 -1.48190 -1.96157 -1.88309 -1.26879 -0.29346 0.76537 1.60642 1.99037 1.80798 1.11114 0.09814 -0.94279 -1.71546 -2.00000 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 1.66294 0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114 0.00000 1.11114 1.84776 1.96157 1.41421 0.39018 -0.76537 -1.66294 -2.00000 -1.66294 -0.76537 0.39018 1.41421 1.96157 1.84776 1.11114 0.00000 -1.11114 -1.84776 -1.96157 -1.41421 -0.39018 0.76537 1.66294 2.00000 1.66294 0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114 0.00000 1.11114 1.84776 1.96157 1.41421 0.39018 -0.76537 -1.66294 -2.00000 -1.66294 -0.76537 0.39018 1.41421 1.96157 1.84776 1.11114 0.00000 -1.11114 -1.84776 -1.96157 -1.41421 -0.39018 0.76537 1.66294 2.00000 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 1.60642 0.58057 -0.67378 -1.66294 -1.99759 -1.54602 -0.48596 0.76537 1.71546 1.99037 1.48190 0.39018 -0.85511 -1.76384 -1.97835 -1.41421 -0.29346 0.94279 1.80798 1.96157 1.34312 0.19603 -1.02821 -1.84776 -1.94006 -1.26879 -0.09814 1.11114 1.88309 1.91388 1.19140 0.00000 -1.19140 -1.91388 -1.88309 -1.11114 0.09814 1.26879 1.94006 1.84776 1.02821 -0.19603 -1.34312 -1.96157 -1.80798 -0.94279 0.29346 1.41421 1.97835 1.76384 0.85511 -0.39018 -1.48190 -1.99037 -1.71546 -0.76537 0.48596 1.54602 1.99759 1.66294 0.67378 -0.58057 -1.60642 -2.00000 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 1.54602 0.39018 -0.94279 -1.84776 -1.91388 -1.11114 0.19603 1.41421 1.99037 1.66294 0.58057 -0.76537 -1.76384 -1.96157 -1.26879 0.00000 1.26879 1.96157 1.76384 0.76537 -0.58057 -1.66294 -1.99037 -1.41421 -0.19603 1.11114 1.91388 1.84776 0.94279 -0.39018 -1.54602 -2.00000 -1.54602 -0.39018 0.94279 1.84776 1.91388 1.11114 -0.19603 -1.41421 -1.99037 -1.66294 -0.58057 0.76537 1.76384 1.96157 1.26879 0.00000 -1.26879 -1.96157 -1.76384 -0.76537 0.58057 1.66294 1.99037 1.41421 0.19603 -1.11114 -1.91388 -1.84776 -0.94279 0.39018 1.54602 2.00000 ... ... ..... ....... ......... ........... ............. 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E28 * 2.00000 0.39018 -1.84776 -1.11114 1.41421 1.66294 -0.76537 -1.96157 0.00000 1.96157 0.76537 -1.66294 -1.41421 1.11114 1.84776 -0.39018 -2.00000 -0.39018 1.84776 1.11114 -1.41421 -1.66294 0.76537 1.96157 0.00000 -1.96157 -0.76537 1.66294 1.41421 -1.11114 -1.84776 0.39018 2.00000 0.39018 -1.84776 -1.11114 1.41421 1.66294 -0.76537 -1.96157 0.00000 1.96157 0.76537 -1.66294 -1.41421 1.11114 1.84776 -0.39018 -2.00000 -0.39018 1.84776 1.11114 -1.41421 -1.66294 0.76537 1.96157 0.00000 -1.96157 -0.76537 1.66294 1.41421 -1.11114 -1.84776 0.39018 2.00000 ... ... ..... ....... ......... ........... ............. E29 * 2.00000 0.29346 -1.91388 -0.85511 1.66294 1.34312 -1.26879 -1.71546 0.76537 1.94006 -0.19603 -1.99759 -0.39018 1.88309 0.94279 -1.60642 -1.41421 1.19140 1.76384 -0.67378 -1.96157 0.09814 1.99037 0.48596 -1.84776 -1.02821 1.54602 1.48190 -1.11114 -1.80798 0.58057 1.97835 0.00000 -1.97835 -0.58057 1.80798 1.11114 -1.48190 -1.54602 1.02821 1.84776 -0.48596 -1.99037 -0.09814 1.96157 0.67378 -1.76384 -1.19140 1.41421 1.60642 -0.94279 -1.88309 0.39018 1.99759 0.19603 -1.94006 -0.76537 1.71546 1.26879 -1.34312 -1.66294 0.85511 1.91388 -0.29346 -2.00000 ... ... ..... ....... ......... ........... ............. E30 * 2.00000 0.19603 -1.96157 -0.58057 1.84776 0.94279 -1.66294 -1.26879 1.41421 1.54602 -1.11114 -1.76384 0.76537 1.91388 -0.39018 -1.99037 0.00000 1.99037 0.39018 -1.91388 -0.76537 1.76384 1.11114 -1.54602 -1.41421 1.26879 1.66294 -0.94279 -1.84776 0.58057 1.96157 -0.19603 -2.00000 -0.19603 1.96157 0.58057 -1.84776 -0.94279 1.66294 1.26879 -1.41421 -1.54602 1.11114 1.76384 -0.76537 -1.91388 0.39018 1.99037 0.00000 -1.99037 -0.39018 1.91388 0.76537 -1.76384 -1.11114 1.54602 1.41421 -1.26879 -1.66294 0.94279 1.84776 -0.58057 -1.96157 0.19603 2.00000 ... ... ..... ....... ......... ........... ............. 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E34 * 2.00000 -0.19603 -1.96157 0.58057 1.84776 -0.94279 -1.66294 1.26879 1.41421 -1.54602 -1.11114 1.76384 0.76537 -1.91388 -0.39018 1.99037 0.00000 -1.99037 0.39018 1.91388 -0.76537 -1.76384 1.11114 1.54602 -1.41421 -1.26879 1.66294 0.94279 -1.84776 -0.58057 1.96157 0.19603 -2.00000 0.19603 1.96157 -0.58057 -1.84776 0.94279 1.66294 -1.26879 -1.41421 1.54602 1.11114 -1.76384 -0.76537 1.91388 0.39018 -1.99037 0.00000 1.99037 -0.39018 -1.91388 0.76537 1.76384 -1.11114 -1.54602 1.41421 1.26879 -1.66294 -0.94279 1.84776 0.58057 -1.96157 -0.19603 2.00000 ... ... ..... ....... ......... ........... ............. 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E37 * 2.00000 -0.48596 -1.76384 1.34312 1.11114 -1.88309 -0.19603 1.97835 -0.76537 -1.60642 1.54602 0.85511 -1.96157 0.09814 1.91388 -1.02821 -1.41421 1.71546 0.58057 -1.99759 0.39018 1.80798 -1.26879 -1.19140 1.84776 0.29346 -1.99037 0.67378 1.66294 -1.48190 -0.94279 1.94006 0.00000 -1.94006 0.94279 1.48190 -1.66294 -0.67378 1.99037 -0.29346 -1.84776 1.19140 1.26879 -1.80798 -0.39018 1.99759 -0.58057 -1.71546 1.41421 1.02821 -1.91388 -0.09814 1.96157 -0.85511 -1.54602 1.60642 0.76537 -1.97835 0.19603 1.88309 -1.11114 -1.34312 1.76384 0.48596 -2.00000 ... ... ..... ....... ......... ........... ............. 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E48 * 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 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 ... ... ..... ....... ......... ........... ............. E49 * 2.00000 -1.48190 0.19603 1.19140 -1.96157 1.71546 -0.58057 -0.85511 1.84776 -1.88309 0.94279 0.48596 -1.66294 1.97835 -1.26879 -0.09814 1.41421 -1.99759 1.54602 -0.29346 -1.11114 1.94006 -1.76384 0.67378 0.76537 -1.80798 1.91388 -1.02821 -0.39018 1.60642 -1.99037 1.34312 0.00000 -1.34312 1.99037 -1.60642 0.39018 1.02821 -1.91388 1.80798 -0.76537 -0.67378 1.76384 -1.94006 1.11114 0.29346 -1.54602 1.99759 -1.41421 0.09814 1.26879 -1.97835 1.66294 -0.48596 -0.94279 1.88309 -1.84776 0.85511 0.58057 -1.71546 1.96157 -1.19140 -0.19603 1.48190 -2.00000 ... ... ..... ....... ......... ........... ............. E50 * 2.00000 -1.54602 0.39018 0.94279 -1.84776 1.91388 -1.11114 -0.19603 1.41421 -1.99037 1.66294 -0.58057 -0.76537 1.76384 -1.96157 1.26879 0.00000 -1.26879 1.96157 -1.76384 0.76537 0.58057 -1.66294 1.99037 -1.41421 0.19603 1.11114 -1.91388 1.84776 -0.94279 -0.39018 1.54602 -2.00000 1.54602 -0.39018 -0.94279 1.84776 -1.91388 1.11114 0.19603 -1.41421 1.99037 -1.66294 0.58057 0.76537 -1.76384 1.96157 -1.26879 0.00000 1.26879 -1.96157 1.76384 -0.76537 -0.58057 1.66294 -1.99037 1.41421 -0.19603 -1.11114 1.91388 -1.84776 0.94279 0.39018 -1.54602 2.00000 ... ... ..... ....... ......... ........... ............. E51 * 2.00000 -1.60642 0.58057 0.67378 -1.66294 1.99759 -1.54602 0.48596 0.76537 -1.71546 1.99037 -1.48190 0.39018 0.85511 -1.76384 1.97835 -1.41421 0.29346 0.94279 -1.80798 1.96157 -1.34312 0.19603 1.02821 -1.84776 1.94006 -1.26879 0.09814 1.11114 -1.88309 1.91388 -1.19140 0.00000 1.19140 -1.91388 1.88309 -1.11114 -0.09814 1.26879 -1.94006 1.84776 -1.02821 -0.19603 1.34312 -1.96157 1.80798 -0.94279 -0.29346 1.41421 -1.97835 1.76384 -0.85511 -0.39018 1.48190 -1.99037 1.71546 -0.76537 -0.48596 1.54602 -1.99759 1.66294 -0.67378 -0.58057 1.60642 -2.00000 ... ... ..... ....... ......... ........... ............. E52 * 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 1.66294 -0.76537 -0.39018 1.41421 -1.96157 1.84776 -1.11114 0.00000 1.11114 -1.84776 1.96157 -1.41421 0.39018 0.76537 -1.66294 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 1.66294 -0.76537 -0.39018 1.41421 -1.96157 1.84776 -1.11114 0.00000 1.11114 -1.84776 1.96157 -1.41421 0.39018 0.76537 -1.66294 2.00000 ... ... ..... ....... ......... ........... ............. E53 * 2.00000 -1.71546 0.94279 0.09814 -1.11114 1.80798 -1.99037 1.60642 -0.76537 -0.29346 1.26879 -1.88309 1.96157 -1.48190 0.58057 0.48596 -1.41421 1.94006 -1.91388 1.34312 -0.39018 -0.67378 1.54602 -1.97835 1.84776 -1.19140 0.19603 0.85511 -1.66294 1.99759 -1.76384 1.02821 0.00000 -1.02821 1.76384 -1.99759 1.66294 -0.85511 -0.19603 1.19140 -1.84776 1.97835 -1.54602 0.67378 0.39018 -1.34312 1.91388 -1.94006 1.41421 -0.48596 -0.58057 1.48190 -1.96157 1.88309 -1.26879 0.29346 0.76537 -1.60642 1.99037 -1.80798 1.11114 -0.09814 -0.94279 1.71546 -2.00000 ... ... ..... ....... ......... ........... ............. E54 * 2.00000 -1.76384 1.11114 -0.19603 -0.76537 1.54602 -1.96157 1.91388 -1.41421 0.58057 0.39018 -1.26879 1.84776 -1.99037 1.66294 -0.94279 0.00000 0.94279 -1.66294 1.99037 -1.84776 1.26879 -0.39018 -0.58057 1.41421 -1.91388 1.96157 -1.54602 0.76537 0.19603 -1.11114 1.76384 -2.00000 1.76384 -1.11114 0.19603 0.76537 -1.54602 1.96157 -1.91388 1.41421 -0.58057 -0.39018 1.26879 -1.84776 1.99037 -1.66294 0.94279 0.00000 -0.94279 1.66294 -1.99037 1.84776 -1.26879 0.39018 0.58057 -1.41421 1.91388 -1.96157 1.54602 -0.76537 -0.19603 1.11114 -1.76384 2.00000 ... ... ..... ....... ......... ........... ............. E55 * 2.00000 -1.80798 1.26879 -0.48596 -0.39018 1.19140 -1.76384 1.99759 -1.84776 1.34312 -0.58057 -0.29346 1.11114 -1.71546 1.99037 -1.88309 1.41421 -0.67378 -0.19603 1.02821 -1.66294 1.97835 -1.91388 1.48190 -0.76537 -0.09814 0.94279 -1.60642 1.96157 -1.94006 1.54602 -0.85511 0.00000 0.85511 -1.54602 1.94006 -1.96157 1.60642 -0.94279 0.09814 0.76537 -1.48190 1.91388 -1.97835 1.66294 -1.02821 0.19603 0.67378 -1.41421 1.88309 -1.99037 1.71546 -1.11114 0.29346 0.58057 -1.34312 1.84776 -1.99759 1.76384 -1.19140 0.39018 0.48596 -1.26879 1.80798 -2.00000 ... ... ..... ....... ......... ........... ............. E56 * 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 ... ... ..... ....... ......... ........... ............. E57 * 2.00000 -1.88309 1.54602 -1.02821 0.39018 0.29346 -0.94279 1.48190 -1.84776 1.99759 -1.91388 1.60642 -1.11114 0.48596 0.19603 -0.85511 1.41421 -1.80798 1.99037 -1.94006 1.66294 -1.19140 0.58057 0.09814 -0.76537 1.34312 -1.76384 1.97835 -1.96157 1.71546 -1.26879 0.67378 0.00000 -0.67378 1.26879 -1.71546 1.96157 -1.97835 1.76384 -1.34312 0.76537 -0.09814 -0.58057 1.19140 -1.66294 1.94006 -1.99037 1.80798 -1.41421 0.85511 -0.19603 -0.48596 1.11114 -1.60642 1.91388 -1.99759 1.84776 -1.48190 0.94279 -0.29346 -0.39018 1.02821 -1.54602 1.88309 -2.00000 ... ... ..... ....... ......... ........... ............. E58 * 2.00000 -1.91388 1.66294 -1.26879 0.76537 -0.19603 -0.39018 0.94279 -1.41421 1.76384 -1.96157 1.99037 -1.84776 1.54602 -1.11114 0.58057 0.00000 -0.58057 1.11114 -1.54602 1.84776 -1.99037 1.96157 -1.76384 1.41421 -0.94279 0.39018 0.19603 -0.76537 1.26879 -1.66294 1.91388 -2.00000 1.91388 -1.66294 1.26879 -0.76537 0.19603 0.39018 -0.94279 1.41421 -1.76384 1.96157 -1.99037 1.84776 -1.54602 1.11114 -0.58057 0.00000 0.58057 -1.11114 1.54602 -1.84776 1.99037 -1.96157 1.76384 -1.41421 0.94279 -0.39018 -0.19603 0.76537 -1.26879 1.66294 -1.91388 2.00000 ... ... ..... ....... ......... ........... ............. E59 * 2.00000 -1.94006 1.76384 -1.48190 1.11114 -0.67378 0.19603 0.29346 -0.76537 1.19140 -1.54602 1.80798 -1.96157 1.99759 -1.91388 1.71546 -1.41421 1.02821 -0.58057 0.09814 0.39018 -0.85511 1.26879 -1.60642 1.84776 -1.97835 1.99037 -1.88309 1.66294 -1.34312 0.94279 -0.48596 0.00000 0.48596 -0.94279 1.34312 -1.66294 1.88309 -1.99037 1.97835 -1.84776 1.60642 -1.26879 0.85511 -0.39018 -0.09814 0.58057 -1.02821 1.41421 -1.71546 1.91388 -1.99759 1.96157 -1.80798 1.54602 -1.19140 0.76537 -0.29346 -0.19603 0.67378 -1.11114 1.48190 -1.76384 1.94006 -2.00000 ... ... ..... ....... ......... ........... ............. E60 * 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 1.96157 -1.84776 1.66294 -1.41421 1.11114 -0.76537 0.39018 0.00000 -0.39018 0.76537 -1.11114 1.41421 -1.66294 1.84776 -1.96157 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 1.96157 -1.84776 1.66294 -1.41421 1.11114 -0.76537 0.39018 0.00000 -0.39018 0.76537 -1.11114 1.41421 -1.66294 1.84776 -1.96157 2.00000 ... ... ..... ....... ......... ........... ............. E61 * 2.00000 -1.97835 1.91388 -1.80798 1.66294 -1.48190 1.26879 -1.02821 0.76537 -0.48596 0.19603 0.09814 -0.39018 0.67378 -0.94279 1.19140 -1.41421 1.60642 -1.76384 1.88309 -1.96157 1.99759 -1.99037 1.94006 -1.84776 1.71546 -1.54602 1.34312 -1.11114 0.85511 -0.58057 0.29346 0.00000 -0.29346 0.58057 -0.85511 1.11114 -1.34312 1.54602 -1.71546 1.84776 -1.94006 1.99037 -1.99759 1.96157 -1.88309 1.76384 -1.60642 1.41421 -1.19140 0.94279 -0.67378 0.39018 -0.09814 -0.19603 0.48596 -0.76537 1.02821 -1.26879 1.48190 -1.66294 1.80798 -1.91388 1.97835 -2.00000 ... ... ..... ....... ......... ........... ............. E62 * 2.00000 -1.99037 1.96157 -1.91388 1.84776 -1.76384 1.66294 -1.54602 1.41421 -1.26879 1.11114 -0.94279 0.76537 -0.58057 0.39018 -0.19603 0.00000 0.19603 -0.39018 0.58057 -0.76537 0.94279 -1.11114 1.26879 -1.41421 1.54602 -1.66294 1.76384 -1.84776 1.91388 -1.96157 1.99037 -2.00000 1.99037 -1.96157 1.91388 -1.84776 1.76384 -1.66294 1.54602 -1.41421 1.26879 -1.11114 0.94279 -0.76537 0.58057 -0.39018 0.19603 0.00000 -0.19603 0.39018 -0.58057 0.76537 -0.94279 1.11114 -1.26879 1.41421 -1.54602 1.66294 -1.76384 1.84776 -1.91388 1.96157 -1.99037 2.00000 ... ... ..... ....... ......... ........... ............. E63 * 2.00000 -1.99759 1.99037 -1.97835 1.96157 -1.94006 1.91388 -1.88309 1.84776 -1.80798 1.76384 -1.71546 1.66294 -1.60642 1.54602 -1.48190 1.41421 -1.34312 1.26879 -1.19140 1.11114 -1.02821 0.94279 -0.85511 0.76537 -0.67378 0.58057 -0.48596 0.39018 -0.29346 0.19603 -0.09814 0.00000 0.09814 -0.19603 0.29346 -0.39018 0.48596 -0.58057 0.67378 -0.76537 0.85511 -0.94279 1.02821 -1.11114 1.19140 -1.26879 1.34312 -1.41421 1.48190 -1.54602 1.60642 -1.66294 1.71546 -1.76384 1.80798 -1.84776 1.88309 -1.91388 1.94006 -1.96157 1.97835 -1.99037 1.99759 -2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.997590912410 = 2*cos(2*π/128) = 2*cos(π/64) = √2+√2+√2+√2+√2 1.990369453344 = 2*cos(4*π/128) = 2*cos(π/32) = √2+√2+√2+√2 1.978353019930 = 2*cos(6*π/128) = 2*cos(3*π/64) = √2+√2+√2+√2−√2 1.961570560806 = 2*cos(8*π/128) = 2*cos(π/16) = √2+√2+√2 1.940062506389 = 2*cos(10*π/128) = 2*cos(5*π/64) = √2+√2+√2−√2−√2 1.913880671464 = 2*cos(12*π/128) = 2*cos(3*π/32) = √2+√2+√2−√2 1.883088130366 = 2*cos(14*π/128) = 2*cos(7*π/64) = √2+√2+√2−√2+√2 1.847759065023 = 2*cos(16*π/128) = 2*cos(π/8) = √2+√2 1.807978586247 = 2*cos(18*π/128) = 2*cos(9*π/64) = √2+√2−√2−√2+√2 1.763842528697 = 2*cos(20*π/128) = 2*cos(5*π/32) = √2+√2−√2−√2 1.715457220001 = 2*cos(22*π/128) = 2*cos(11*π/64) = √2+√2−√2−√2−√2 1.662939224605 = 2*cos(24*π/128) = 2*cos(3*π/16) = √2+√2−√2 1.606415062961 = 2*cos(26*π/128) = 2*cos(13*π/64) = √2+√2−√2+√2−√2 1.546020906725 = 2*cos(28*π/128) = 2*cos(7*π/32) = √2+√2−√2+√2 1.481902250710 = 2*cos(30*π/128) = 2*cos(15*π/64) = √2+√2−√2+√2+√2 1.414213562373 = 2*cos(32*π/128) = 2*cos(π/4) = √2 1.343117909694 = 2*cos(34*π/128) = 2*cos(17*π/64) = √2−√2−√2+√2+√2 1.268786568327 = 2*cos(36*π/128) = 2*cos(9*π/32) = √2−√2−√2+√2 1.191398608985 = 2*cos(38*π/128) = 2*cos(19*π/64) = √2−√2−√2+√2−√2 1.111140466039 = 2*cos(40*π/128) = 2*cos(5*π/16) = √2−√2−√2 1.028205488386 = 2*cos(42*π/128) = 2*cos(21*π/64) = √2−√2−√2−√2−√2 0.942793473652 = 2*cos(44*π/128) = 2*cos(11*π/32) = √2−√2−√2−√2 0.855110186861 = 2*cos(46*π/128) = 2*cos(23*π/64) = √2−√2−√2−√2+√2 0.765366864730 = 2*cos(48*π/128) = 2*cos(3*π/8) = √2−√2 0.673779706784 = 2*cos(50*π/128) = 2*cos(25*π/64) = √2−√2+√2−√2+√2 0.580569354509 = 2*cos(52*π/128) = 2*cos(13*π/32) = √2−√2+√2−√2 0.485960359807 = 2*cos(54*π/128) = 2*cos(27*π/64) = √2−√2+√2−√2−√2 0.390180644032 = 2*cos(56*π/128) = 2*cos(7*π/16) = √2−√2+√2 0.293460948911 = 2*cos(58*π/128) = 2*cos(29*π/64) = √2−√2+√2+√2−√2 0.196034280659 = 2*cos(60*π/128) = 2*cos(15*π/32) = √2−√2+√2+√2 0.098135348655 = 2*cos(62*π/128) = 2*cos(31*π/64) = √2−√2+√2+√2+√2 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 C128 point group is 128, and the order of the principal axis (C128) is 128. The group has 65 irreducible representations. β The C128 point group is isomorphic to S128. γ The C128 point group is generated by one single symmetry element, C128. Therefore, it is a cyclic group. δ The lowest nonvanishing multipole moment in C128 is 2 (dipole moment). ε This is an Abelian point group (the commutative law holds between all symmetry operations). The C128 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 126 cases have been combined into 63 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 63 pairs of left and right rotations have been combined into one two-membered pseudo-class each. η The 63 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. κ The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.
C126 | ||
C127 | ||
C128 | C128v C128h D128 D128h D128d S128 | |
C136 | ||
C144 |
This Character Table for the C128 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.