sage: H = DirichletGroup(185)
pari: g = idealstar(,185,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 144 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{4}\times C_{36}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{185}(112,\cdot)$, $\chi_{185}(76,\cdot)$ |
First 32 of 144 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{185}(1,\cdot)\) | 185.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{185}(2,\cdot)\) | 185.bc | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(-i\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{185}(3,\cdot)\) | 185.y | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(-1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |
\(\chi_{185}(4,\cdot)\) | 185.v | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(-1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{185}(6,\cdot)\) | 185.g | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(-1\) | \(i\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(1\) | \(i\) |
\(\chi_{185}(7,\cdot)\) | 185.bd | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
\(\chi_{185}(8,\cdot)\) | 185.u | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{185}(9,\cdot)\) | 185.x | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{185}(11,\cdot)\) | 185.m | 6 | no | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{185}(12,\cdot)\) | 185.bd | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
\(\chi_{185}(13,\cdot)\) | 185.bc | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(i\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{185}(14,\cdot)\) | 185.q | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) |
\(\chi_{185}(16,\cdot)\) | 185.o | 9 | no | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{185}(17,\cdot)\) | 185.z | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(i\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{185}(18,\cdot)\) | 185.z | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-i\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{185}(19,\cdot)\) | 185.ba | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) |
\(\chi_{185}(21,\cdot)\) | 185.w | 18 | no | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(-1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{185}(22,\cdot)\) | 185.z | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(i\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{185}(23,\cdot)\) | 185.u | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{185}(24,\cdot)\) | 185.ba | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(-i\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) |
\(\chi_{185}(26,\cdot)\) | 185.e | 3 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{185}(27,\cdot)\) | 185.r | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
\(\chi_{185}(28,\cdot)\) | 185.y | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(-1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
\(\chi_{185}(29,\cdot)\) | 185.q | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{185}(31,\cdot)\) | 185.g | 4 | no | \(-1\) | \(1\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(1\) | \(-i\) | \(1\) | \(-1\) | \(1\) | \(-i\) |
\(\chi_{185}(32,\cdot)\) | 185.bc | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(-i\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{185}(33,\cdot)\) | 185.bd | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
\(\chi_{185}(34,\cdot)\) | 185.x | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{185}(36,\cdot)\) | 185.c | 2 | no | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) |
\(\chi_{185}(38,\cdot)\) | 185.i | 4 | no | \(-1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(1\) | \(-i\) | \(i\) |
\(\chi_{185}(39,\cdot)\) | 185.ba | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(-i\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) |
\(\chi_{185}(41,\cdot)\) | 185.w | 18 | no | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(-1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) |