sage: H = DirichletGroup(720)
pari: g = idealstar(,720,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 192 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{4}\times C_{12}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{720}(271,\cdot)$, $\chi_{720}(181,\cdot)$, $\chi_{720}(641,\cdot)$, $\chi_{720}(577,\cdot)$ |
First 32 of 192 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{720}(1,\cdot)\) | 720.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{720}(7,\cdot)\) | 720.ci | 12 | no | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(11,\cdot)\) | 720.cf | 12 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(13,\cdot)\) | 720.cr | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{720}(17,\cdot)\) | 720.w | 4 | no | \(1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(-1\) |
\(\chi_{720}(19,\cdot)\) | 720.r | 4 | no | \(-1\) | \(1\) | \(-1\) | \(i\) | \(-i\) | \(-1\) | \(-i\) | \(-1\) | \(i\) | \(-1\) | \(i\) | \(-1\) |
\(\chi_{720}(23,\cdot)\) | 720.cv | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{720}(29,\cdot)\) | 720.ch | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(31,\cdot)\) | 720.cb | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{720}(37,\cdot)\) | 720.bb | 4 | no | \(-1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(i\) | \(i\) | \(i\) | \(1\) | \(-1\) | \(-1\) |
\(\chi_{720}(41,\cdot)\) | 720.bq | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{720}(43,\cdot)\) | 720.cp | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{720}(47,\cdot)\) | 720.ck | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{720}(49,\cdot)\) | 720.by | 6 | no | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{720}(53,\cdot)\) | 720.bg | 4 | no | \(1\) | \(1\) | \(i\) | \(-i\) | \(1\) | \(i\) | \(i\) | \(i\) | \(-i\) | \(1\) | \(1\) | \(1\) |
\(\chi_{720}(59,\cdot)\) | 720.da | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(1\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{720}(61,\cdot)\) | 720.db | 12 | no | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{720}(67,\cdot)\) | 720.cp | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{720}(71,\cdot)\) | 720.b | 2 | no | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
\(\chi_{720}(73,\cdot)\) | 720.y | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) | \(i\) | \(1\) | \(1\) | \(i\) | \(1\) |
\(\chi_{720}(77,\cdot)\) | 720.cm | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{720}(79,\cdot)\) | 720.bu | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(83,\cdot)\) | 720.cs | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(89,\cdot)\) | 720.i | 2 | no | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) |
\(\chi_{720}(91,\cdot)\) | 720.bo | 4 | no | \(-1\) | \(1\) | \(1\) | \(-i\) | \(-i\) | \(1\) | \(i\) | \(1\) | \(-i\) | \(-1\) | \(i\) | \(-1\) |
\(\chi_{720}(97,\cdot)\) | 720.cj | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(101,\cdot)\) | 720.cy | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{720}(103,\cdot)\) | 720.ci | 12 | no | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{720}(107,\cdot)\) | 720.ba | 4 | no | \(-1\) | \(1\) | \(i\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(i\) | \(-i\) | \(-1\) | \(-1\) | \(1\) |
\(\chi_{720}(109,\cdot)\) | 720.bm | 4 | no | \(1\) | \(1\) | \(1\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(1\) | \(i\) | \(1\) | \(i\) | \(-1\) |
\(\chi_{720}(113,\cdot)\) | 720.cu | 12 | no | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{720}(119,\cdot)\) | 720.bt | 6 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |