sage: H = DirichletGroup(864)
pari: g = idealstar(,864,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 288 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{72}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{864}(703,\cdot)$, $\chi_{864}(325,\cdot)$, $\chi_{864}(353,\cdot)$ |
First 32 of 288 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{864}(1,\cdot)\) | 864.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{864}(5,\cdot)\) | 864.bs | 72 | yes | \(-1\) | \(1\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{864}(7,\cdot)\) | 864.bp | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{864}(11,\cdot)\) | 864.bt | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{864}(13,\cdot)\) | 864.bu | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{864}(17,\cdot)\) | 864.n | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{864}(19,\cdot)\) | 864.bl | 24 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{864}(23,\cdot)\) | 864.br | 36 | no | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{864}(25,\cdot)\) | 864.bo | 36 | no | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{864}(29,\cdot)\) | 864.bs | 72 | yes | \(-1\) | \(1\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{864}(31,\cdot)\) | 864.be | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{864}(35,\cdot)\) | 864.bn | 24 | no | \(1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{864}(37,\cdot)\) | 864.bk | 24 | no | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{864}(41,\cdot)\) | 864.bq | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{864}(43,\cdot)\) | 864.bv | 72 | yes | \(-1\) | \(1\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{864}(47,\cdot)\) | 864.bh | 18 | no | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{864}(49,\cdot)\) | 864.bf | 18 | no | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{864}(53,\cdot)\) | 864.x | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(1\) |
\(\chi_{864}(55,\cdot)\) | 864.m | 4 | no | \(-1\) | \(1\) | \(-i\) | \(1\) | \(i\) | \(i\) | \(1\) | \(-i\) | \(1\) | \(-1\) | \(i\) | \(-1\) |
\(\chi_{864}(59,\cdot)\) | 864.bt | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{864}(61,\cdot)\) | 864.bu | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{864}(65,\cdot)\) | 864.bg | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{864}(67,\cdot)\) | 864.bv | 72 | yes | \(-1\) | \(1\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{864}(71,\cdot)\) | 864.z | 12 | no | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{864}(73,\cdot)\) | 864.bc | 12 | no | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{864}(77,\cdot)\) | 864.bs | 72 | yes | \(-1\) | \(1\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{864}(79,\cdot)\) | 864.bd | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{864}(83,\cdot)\) | 864.bt | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{864}(85,\cdot)\) | 864.bu | 72 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{864}(89,\cdot)\) | 864.bb | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{864}(91,\cdot)\) | 864.bl | 24 | no | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{864}(95,\cdot)\) | 864.bi | 18 | no | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) |