sage: H = DirichletGroup(51)
pari: g = idealstar(,51,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 32 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{16}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{51}(35,\cdot)$, $\chi_{51}(37,\cdot)$ |
First 32 of 32 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{51}(1,\cdot)\) | 51.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{51}(2,\cdot)\) | 51.g | 8 | yes | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(1\) |
\(\chi_{51}(4,\cdot)\) | 51.e | 4 | no | \(1\) | \(1\) | \(-1\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(i\) | \(i\) | \(1\) | \(-i\) | \(1\) |
\(\chi_{51}(5,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(-1\) |
\(\chi_{51}(7,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(-1\) |
\(\chi_{51}(8,\cdot)\) | 51.g | 8 | yes | \(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
\(\chi_{51}(10,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
\(\chi_{51}(11,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) |
\(\chi_{51}(13,\cdot)\) | 51.e | 4 | no | \(1\) | \(1\) | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) | \(i\) | \(1\) |
\(\chi_{51}(14,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) |
\(\chi_{51}(16,\cdot)\) | 51.d | 2 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
\(\chi_{51}(19,\cdot)\) | 51.h | 8 | no | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) |
\(\chi_{51}(20,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(-1\) |
\(\chi_{51}(22,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
\(\chi_{51}(23,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) |
\(\chi_{51}(25,\cdot)\) | 51.h | 8 | no | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(1\) |
\(\chi_{51}(26,\cdot)\) | 51.g | 8 | yes | \(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(1\) |
\(\chi_{51}(28,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) |
\(\chi_{51}(29,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
\(\chi_{51}(31,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(-1\) |
\(\chi_{51}(32,\cdot)\) | 51.g | 8 | yes | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) |
\(\chi_{51}(35,\cdot)\) | 51.b | 2 | no | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
\(\chi_{51}(37,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) |
\(\chi_{51}(38,\cdot)\) | 51.f | 4 | yes | \(-1\) | \(1\) | \(1\) | \(1\) | \(i\) | \(i\) | \(1\) | \(i\) | \(-i\) | \(1\) | \(i\) | \(1\) |
\(\chi_{51}(40,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(-1\) |
\(\chi_{51}(41,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
\(\chi_{51}(43,\cdot)\) | 51.h | 8 | no | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
\(\chi_{51}(44,\cdot)\) | 51.i | 16 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(-1\) |
\(\chi_{51}(46,\cdot)\) | 51.j | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(-1\) |
\(\chi_{51}(47,\cdot)\) | 51.f | 4 | yes | \(-1\) | \(1\) | \(1\) | \(1\) | \(-i\) | \(-i\) | \(1\) | \(-i\) | \(i\) | \(1\) | \(-i\) | \(1\) |
\(\chi_{51}(49,\cdot)\) | 51.h | 8 | no | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(1\) |
\(\chi_{51}(50,\cdot)\) | 51.c | 2 | yes | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(1\) |