sage: H = DirichletGroup(432)
pari: g = idealstar(,432,2)
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 | = | $\chi_{432}(271,\cdot)$, $\chi_{432}(325,\cdot)$, $\chi_{432}(353,\cdot)$ |
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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{432}(1,\cdot)\) | 432.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{432}(5,\cdot)\) | 432.bi | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{432}(7,\cdot)\) | 432.z | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{432}(11,\cdot)\) | 432.bj | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{432}(13,\cdot)\) | 432.bg | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{432}(17,\cdot)\) | 432.q | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{432}(19,\cdot)\) | 432.w | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{432}(23,\cdot)\) | 432.bd | 18 | no | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{432}(25,\cdot)\) | 432.bb | 18 | no | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{432}(29,\cdot)\) | 432.bi | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{432}(31,\cdot)\) | 432.ba | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{432}(35,\cdot)\) | 432.v | 12 | no | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{432}(37,\cdot)\) | 432.y | 12 | no | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{432}(41,\cdot)\) | 432.bf | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{432}(43,\cdot)\) | 432.bh | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{432}(47,\cdot)\) | 432.be | 18 | no | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{432}(49,\cdot)\) | 432.u | 9 | no | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{432}(53,\cdot)\) | 432.j | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(-i\) | \(1\) | \(-1\) | \(i\) | \(1\) |
\(\chi_{432}(55,\cdot)\) | 432.b | 2 | no | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
\(\chi_{432}(59,\cdot)\) | 432.bj | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{432}(61,\cdot)\) | 432.bg | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{432}(65,\cdot)\) | 432.bc | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{432}(67,\cdot)\) | 432.bh | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{432}(71,\cdot)\) | 432.p | 6 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{432}(73,\cdot)\) | 432.r | 6 | no | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{432}(77,\cdot)\) | 432.bi | 36 | yes | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{432}(79,\cdot)\) | 432.ba | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{432}(83,\cdot)\) | 432.bj | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{432}(85,\cdot)\) | 432.bg | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{432}(89,\cdot)\) | 432.n | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{432}(91,\cdot)\) | 432.w | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{432}(95,\cdot)\) | 432.be | 18 | no | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) |