sage: H = DirichletGroup(163)
pari: g = idealstar(,163,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 162 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{162}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{163}(2,\cdot)$ |
First 32 of 162 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.
Character | Orbit | Order | Primitive | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{163}(1,\cdot)\) | 163.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{163}(2,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{162}\right)\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{47}{162}\right)\) |
\(\chi_{163}(3,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{83}{162}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{49}{162}\right)\) |
\(\chi_{163}(4,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) |
\(\chi_{163}(5,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) |
\(\chi_{163}(6,\cdot)\) | 163.g | 27 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) |
\(\chi_{163}(7,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{83}{162}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{29}{162}\right)\) |
\(\chi_{163}(8,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) |
\(\chi_{163}(9,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{49}{81}\right)\) |
\(\chi_{163}(10,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{77}{81}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) |
\(\chi_{163}(11,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{103}{162}\right)\) |
\(\chi_{163}(12,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{103}{162}\right)\) | \(e\left(\frac{35}{162}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{67}{162}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{143}{162}\right)\) |
\(\chi_{163}(13,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) |
\(\chi_{163}(14,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{38}{81}\right)\) |
\(\chi_{163}(15,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{53}{81}\right)\) |
\(\chi_{163}(16,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) |
\(\chi_{163}(17,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) |
\(\chi_{163}(18,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{145}{162}\right)\) |
\(\chi_{163}(19,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{151}{162}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{43}{162}\right)\) |
\(\chi_{163}(20,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{97}{162}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{151}{162}\right)\) |
\(\chi_{163}(21,\cdot)\) | 163.g | 27 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) |
\(\chi_{163}(22,\cdot)\) | 163.g | 27 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) |
\(\chi_{163}(23,\cdot)\) | 163.f | 18 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{163}(24,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) |
\(\chi_{163}(25,\cdot)\) | 163.g | 27 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) |
\(\chi_{163}(26,\cdot)\) | 163.i | 81 | yes | \(1\) | \(1\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) |
\(\chi_{163}(27,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) |
\(\chi_{163}(28,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) |
\(\chi_{163}(29,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{35}{162}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{7}{162}\right)\) |
\(\chi_{163}(30,\cdot)\) | 163.f | 18 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{163}(31,\cdot)\) | 163.h | 54 | yes | \(-1\) | \(1\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) |
\(\chi_{163}(32,\cdot)\) | 163.j | 162 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{162}\right)\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{73}{162}\right)\) |