Properties

 Modulus 193 Structure $$C_{192}$$ Order 192

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(193)
pari: g = idealstar(,193,2)

Character group

 sage: G.order() pari: g.no Order = 192 sage: H.invariants() pari: g.cyc Structure = $$C_{192}$$ sage: H.gens() pari: g.gen Generators = $\chi_{193}(5,\cdot)$

First 32 of 192 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{193}(1,\cdot)$$ 193.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{193}(2,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{17}{96}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{19}{96}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{193}(3,\cdot)$$ 193.h 16 Yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{193}(4,\cdot)$$ 193.k 48 Yes $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{193}(5,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{96}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{1}{192}\right)$$ $$e\left(\frac{59}{96}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{35}{192}\right)$$ $$e\left(\frac{61}{64}\right)$$
$$\chi_{193}(6,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{59}{96}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{49}{96}\right)$$ $$e\left(\frac{15}{32}\right)$$
$$\chi_{193}(7,\cdot)$$ 193.i 24 Yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{193}(8,\cdot)$$ 193.j 32 Yes $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$
$$\chi_{193}(9,\cdot)$$ 193.f 8 Yes $$1$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$1$$ $$i$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{193}(10,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{19}{96}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{35}{192}\right)$$ $$e\left(\frac{49}{96}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{73}{192}\right)$$ $$e\left(\frac{23}{64}\right)$$
$$\chi_{193}(11,\cdot)$$ 193.l 64 Yes $$-1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$
$$\chi_{193}(12,\cdot)$$ 193.i 24 Yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{193}(13,\cdot)$$ 193.l 64 Yes $$-1$$ $$1$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{25}{64}\right)$$
$$\chi_{193}(14,\cdot)$$ 193.j 32 Yes $$1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$
$$\chi_{193}(15,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{85}{192}\right)$$ $$e\left(\frac{23}{96}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{95}{192}\right)$$ $$e\left(\frac{1}{64}\right)$$
$$\chi_{193}(16,\cdot)$$ 193.i 24 Yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{193}(17,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{47}{96}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{31}{192}\right)$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{125}{192}\right)$$ $$e\left(\frac{35}{64}\right)$$
$$\chi_{193}(18,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{79}{96}\right)$$ $$e\left(\frac{17}{32}\right)$$
$$\chi_{193}(19,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{65}{96}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{145}{192}\right)$$ $$e\left(\frac{11}{96}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{83}{192}\right)$$ $$e\left(\frac{13}{64}\right)$$
$$\chi_{193}(20,\cdot)$$ 193.l 64 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{49}{64}\right)$$
$$\chi_{193}(21,\cdot)$$ 193.k 48 Yes $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{193}(22,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{41}{96}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{25}{192}\right)$$ $$e\left(\frac{35}{96}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{107}{192}\right)$$ $$e\left(\frac{53}{64}\right)$$
$$\chi_{193}(23,\cdot)$$ 193.j 32 Yes $$1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{193}(24,\cdot)$$ 193.j 32 Yes $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$
$$\chi_{193}(25,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{96}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{35}{96}\right)$$ $$e\left(\frac{29}{32}\right)$$
$$\chi_{193}(26,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{95}{96}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{175}{192}\right)$$ $$e\left(\frac{53}{96}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{173}{192}\right)$$ $$e\left(\frac{51}{64}\right)$$
$$\chi_{193}(27,\cdot)$$ 193.h 16 Yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{193}(28,\cdot)$$ 193.k 48 Yes $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{193}(29,\cdot)$$ 193.l 64 Yes $$-1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{15}{64}\right)$$
$$\chi_{193}(30,\cdot)$$ 193.n 192 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{96}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{119}{192}\right)$$ $$e\left(\frac{13}{96}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{133}{192}\right)$$ $$e\left(\frac{27}{64}\right)$$
$$\chi_{193}(31,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{41}{96}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{91}{96}\right)$$ $$e\left(\frac{5}{32}\right)$$
$$\chi_{193}(32,\cdot)$$ 193.m 96 Yes $$1$$ $$1$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{85}{96}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{95}{96}\right)$$ $$e\left(\frac{1}{32}\right)$$