sage: H = DirichletGroup(78)
pari: g = idealstar(,78,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 24 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{12}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{78}(53,\cdot)$, $\chi_{78}(67,\cdot)$ |
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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{78}(1,\cdot)\) | 78.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{78}(5,\cdot)\) | 78.g | 4 | no | \(1\) | \(1\) | \(i\) | \(i\) | \(-i\) | \(1\) | \(-i\) | \(1\) | \(-1\) | \(-1\) | \(-i\) | \(-1\) |
\(\chi_{78}(7,\cdot)\) | 78.l | 12 | no | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{78}(11,\cdot)\) | 78.k | 12 | no | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{78}(17,\cdot)\) | 78.j | 6 | no | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{78}(19,\cdot)\) | 78.l | 12 | no | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{78}(23,\cdot)\) | 78.j | 6 | no | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{78}(25,\cdot)\) | 78.b | 2 | no | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
\(\chi_{78}(29,\cdot)\) | 78.h | 6 | no | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{78}(31,\cdot)\) | 78.f | 4 | no | \(-1\) | \(1\) | \(-i\) | \(i\) | \(i\) | \(-1\) | \(-i\) | \(-1\) | \(-1\) | \(1\) | \(-i\) | \(1\) |
\(\chi_{78}(35,\cdot)\) | 78.h | 6 | no | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{78}(37,\cdot)\) | 78.l | 12 | no | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{78}(41,\cdot)\) | 78.k | 12 | no | \(1\) | \(1\) | \(i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{78}(43,\cdot)\) | 78.i | 6 | no | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{78}(47,\cdot)\) | 78.g | 4 | no | \(1\) | \(1\) | \(-i\) | \(-i\) | \(i\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(-1\) | \(i\) | \(-1\) |
\(\chi_{78}(49,\cdot)\) | 78.i | 6 | no | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{78}(53,\cdot)\) | 78.c | 2 | no | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
\(\chi_{78}(55,\cdot)\) | 78.e | 3 | no | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{78}(59,\cdot)\) | 78.k | 12 | no | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{78}(61,\cdot)\) | 78.e | 3 | no | \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{78}(67,\cdot)\) | 78.l | 12 | no | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{78}(71,\cdot)\) | 78.k | 12 | no | \(1\) | \(1\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{78}(73,\cdot)\) | 78.f | 4 | no | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(-1\) | \(-1\) | \(1\) | \(i\) | \(1\) |
\(\chi_{78}(77,\cdot)\) | 78.d | 2 | no | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |