sage: H = DirichletGroup(91)
pari: g = idealstar(,91,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 72 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{6}\times C_{12}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{91}(66,\cdot)$, $\chi_{91}(15,\cdot)$ |
First 32 of 72 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{91}(1,\cdot)\) | 91.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{91}(2,\cdot)\) | 91.x | 12 | yes | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{91}(3,\cdot)\) | 91.m | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(4,\cdot)\) | 91.k | 6 | yes | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{91}(5,\cdot)\) | 91.bb | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(6,\cdot)\) | 91.bc | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(1\) |
\(\chi_{91}(8,\cdot)\) | 91.j | 4 | no | \(-1\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(i\) | \(i\) | \(-i\) | \(1\) | \(-1\) | \(-i\) | \(-1\) |
\(\chi_{91}(9,\cdot)\) | 91.g | 3 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(10,\cdot)\) | 91.p | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(11,\cdot)\) | 91.bd | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(1\) | \(-1\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(12,\cdot)\) | 91.s | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{91}(15,\cdot)\) | 91.y | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) |
\(\chi_{91}(16,\cdot)\) | 91.h | 3 | yes | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(17,\cdot)\) | 91.l | 6 | yes | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(18,\cdot)\) | 91.z | 12 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(19,\cdot)\) | 91.w | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(20,\cdot)\) | 91.bc | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) |
\(\chi_{91}(22,\cdot)\) | 91.f | 3 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) |
\(\chi_{91}(23,\cdot)\) | 91.k | 6 | yes | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(24,\cdot)\) | 91.w | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{91}(25,\cdot)\) | 91.r | 6 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{91}(27,\cdot)\) | 91.d | 2 | no | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
\(\chi_{91}(29,\cdot)\) | 91.f | 3 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) |
\(\chi_{91}(30,\cdot)\) | 91.u | 6 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(31,\cdot)\) | 91.bb | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{91}(32,\cdot)\) | 91.x | 12 | yes | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(33,\cdot)\) | 91.w | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{91}(34,\cdot)\) | 91.i | 4 | yes | \(1\) | \(1\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(-i\) | \(-i\) | \(1\) | \(1\) | \(-i\) | \(1\) |
\(\chi_{91}(36,\cdot)\) | 91.q | 6 | no | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) |
\(\chi_{91}(37,\cdot)\) | 91.x | 12 | yes | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{91}(38,\cdot)\) | 91.s | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{91}(40,\cdot)\) | 91.o | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |