sage: H = DirichletGroup(26)
pari: g = idealstar(,26,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 12 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{12}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{26}(15,\cdot)$ |
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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{26}(1,\cdot)\) | 26.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{26}(3,\cdot)\) | 26.c | 3 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{26}(5,\cdot)\) | 26.d | 4 | no | \(-1\) | \(1\) | \(1\) | \(-i\) | \(i\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-i\) | \(i\) | \(-1\) |
\(\chi_{26}(7,\cdot)\) | 26.f | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{26}(9,\cdot)\) | 26.c | 3 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{26}(11,\cdot)\) | 26.f | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{26}(15,\cdot)\) | 26.f | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{26}(17,\cdot)\) | 26.e | 6 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{26}(19,\cdot)\) | 26.f | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{26}(21,\cdot)\) | 26.d | 4 | no | \(-1\) | \(1\) | \(1\) | \(i\) | \(-i\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(-i\) | \(-1\) |
\(\chi_{26}(23,\cdot)\) | 26.e | 6 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{26}(25,\cdot)\) | 26.b | 2 | no | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) |