sage: H = DirichletGroup(160)
pari: g = idealstar(,160,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 64 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{4}\times C_{8}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{160}(31,\cdot)$, $\chi_{160}(101,\cdot)$, $\chi_{160}(97,\cdot)$ |
First 32 of 64 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{160}(1,\cdot)\) | 160.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{160}(3,\cdot)\) | 160.ba | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{160}(7,\cdot)\) | 160.s | 4 | no | \(1\) | \(1\) | \(1\) | \(i\) | \(1\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(i\) | \(-i\) | \(1\) |
\(\chi_{160}(9,\cdot)\) | 160.q | 4 | no | \(1\) | \(1\) | \(-i\) | \(1\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(1\) | \(i\) |
\(\chi_{160}(11,\cdot)\) | 160.w | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{160}(13,\cdot)\) | 160.v | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{160}(17,\cdot)\) | 160.m | 4 | no | \(-1\) | \(1\) | \(i\) | \(i\) | \(-1\) | \(-1\) | \(i\) | \(i\) | \(1\) | \(-1\) | \(-i\) | \(-i\) |
\(\chi_{160}(19,\cdot)\) | 160.y | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{160}(21,\cdot)\) | 160.x | 8 | no | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{160}(23,\cdot)\) | 160.s | 4 | no | \(1\) | \(1\) | \(1\) | \(-i\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(i\) | \(-i\) | \(i\) | \(1\) |
\(\chi_{160}(27,\cdot)\) | 160.ba | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{160}(29,\cdot)\) | 160.z | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{160}(31,\cdot)\) | 160.b | 2 | no | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
\(\chi_{160}(33,\cdot)\) | 160.p | 4 | no | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(-i\) |
\(\chi_{160}(37,\cdot)\) | 160.v | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{160}(39,\cdot)\) | 160.k | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(-1\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(i\) | \(-1\) | \(i\) |
\(\chi_{160}(41,\cdot)\) | 160.l | 4 | no | \(1\) | \(1\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(i\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-i\) |
\(\chi_{160}(43,\cdot)\) | 160.u | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{160}(47,\cdot)\) | 160.o | 4 | no | \(1\) | \(1\) | \(-i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(i\) | \(-1\) | \(-1\) | \(i\) | \(i\) |
\(\chi_{160}(49,\cdot)\) | 160.f | 2 | no | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
\(\chi_{160}(51,\cdot)\) | 160.w | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{160}(53,\cdot)\) | 160.bb | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{160}(57,\cdot)\) | 160.i | 4 | no | \(-1\) | \(1\) | \(-1\) | \(-i\) | \(1\) | \(i\) | \(-1\) | \(i\) | \(i\) | \(i\) | \(i\) | \(-1\) |
\(\chi_{160}(59,\cdot)\) | 160.y | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{160}(61,\cdot)\) | 160.x | 8 | no | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{160}(63,\cdot)\) | 160.n | 4 | no | \(1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(-1\) | \(i\) | \(-i\) | \(1\) | \(1\) | \(-i\) | \(i\) |
\(\chi_{160}(67,\cdot)\) | 160.u | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{160}(69,\cdot)\) | 160.z | 8 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{160}(71,\cdot)\) | 160.r | 4 | no | \(-1\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(-i\) | \(-i\) | \(1\) | \(i\) | \(i\) | \(1\) | \(-i\) |
\(\chi_{160}(73,\cdot)\) | 160.i | 4 | no | \(-1\) | \(1\) | \(-1\) | \(i\) | \(1\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(-i\) | \(-i\) | \(-1\) |
\(\chi_{160}(77,\cdot)\) | 160.bb | 8 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{160}(79,\cdot)\) | 160.e | 2 | no | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |