sage: H = DirichletGroup(285)
pari: g = idealstar(,285,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 144 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{36}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{285}(191,\cdot)$, $\chi_{285}(172,\cdot)$, $\chi_{285}(211,\cdot)$ |
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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{285}(1,\cdot)\) | 285.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{285}(2,\cdot)\) | 285.bj | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) |
\(\chi_{285}(4,\cdot)\) | 285.be | 18 | no | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{285}(7,\cdot)\) | 285.y | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(-i\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
\(\chi_{285}(8,\cdot)\) | 285.w | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(i\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{285}(11,\cdot)\) | 285.s | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{285}(13,\cdot)\) | 285.bh | 36 | no | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
\(\chi_{285}(14,\cdot)\) | 285.bf | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{285}(16,\cdot)\) | 285.u | 9 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{285}(17,\cdot)\) | 285.bi | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
\(\chi_{285}(22,\cdot)\) | 285.bh | 36 | no | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
\(\chi_{285}(23,\cdot)\) | 285.bi | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |
\(\chi_{285}(26,\cdot)\) | 285.s | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{285}(28,\cdot)\) | 285.bg | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
\(\chi_{285}(29,\cdot)\) | 285.bf | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{285}(31,\cdot)\) | 285.t | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{285}(32,\cdot)\) | 285.bj | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
\(\chi_{285}(34,\cdot)\) | 285.bc | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{285}(37,\cdot)\) | 285.m | 4 | no | \(1\) | \(1\) | \(-i\) | \(-1\) | \(i\) | \(i\) | \(1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(-i\) |
\(\chi_{285}(41,\cdot)\) | 285.z | 18 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{285}(43,\cdot)\) | 285.bg | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
\(\chi_{285}(44,\cdot)\) | 285.bd | 18 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{285}(46,\cdot)\) | 285.t | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{285}(47,\cdot)\) | 285.bi | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
\(\chi_{285}(49,\cdot)\) | 285.r | 6 | no | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{285}(52,\cdot)\) | 285.bh | 36 | no | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
\(\chi_{285}(53,\cdot)\) | 285.bj | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) |
\(\chi_{285}(56,\cdot)\) | 285.h | 2 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) |
\(\chi_{285}(58,\cdot)\) | 285.l | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(-i\) | \(i\) | \(1\) | \(i\) | \(-1\) | \(1\) | \(-i\) | \(-i\) |
\(\chi_{285}(59,\cdot)\) | 285.bf | 18 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{285}(61,\cdot)\) | 285.u | 9 | no | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{285}(62,\cdot)\) | 285.bi | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |