sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(44)
pari: g = idealstar(,44,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 20 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{10}\times C_{2}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{44}(13,\cdot)$, $\chi_{44}(23,\cdot)$ |
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.
orbit label | order | primitive | -1 | 1 | 3 | 5 | 7 | 9 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{44}(1,\cdot)\) | 44.a | 1 | No | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{44}(3,\cdot)\) | 44.h | 10 | Yes | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(-1\) |
\(\chi_{44}(5,\cdot)\) | 44.e | 5 | No | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(1\) |
\(\chi_{44}(7,\cdot)\) | 44.g | 10 | Yes | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-1\) | \(-1\) |
\(\chi_{44}(9,\cdot)\) | 44.e | 5 | No | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(1\) |
\(\chi_{44}(13,\cdot)\) | 44.f | 10 | No | \(-1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) | \(1\) |
\(\chi_{44}(15,\cdot)\) | 44.h | 10 | Yes | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(1\) | \(-1\) |
\(\chi_{44}(17,\cdot)\) | 44.f | 10 | No | \(-1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) | \(1\) |
\(\chi_{44}(19,\cdot)\) | 44.g | 10 | Yes | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(-1\) |
\(\chi_{44}(21,\cdot)\) | 44.d | 2 | No | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
\(\chi_{44}(23,\cdot)\) | 44.b | 2 | No | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
\(\chi_{44}(25,\cdot)\) | 44.e | 5 | No | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(1\) |
\(\chi_{44}(27,\cdot)\) | 44.h | 10 | Yes | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) | \(-1\) |
\(\chi_{44}(29,\cdot)\) | 44.f | 10 | No | \(-1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) | \(1\) |
\(\chi_{44}(31,\cdot)\) | 44.h | 10 | Yes | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(1\) | \(-1\) |
\(\chi_{44}(35,\cdot)\) | 44.g | 10 | Yes | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) | \(-1\) |
\(\chi_{44}(37,\cdot)\) | 44.e | 5 | No | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(1\) |
\(\chi_{44}(39,\cdot)\) | 44.g | 10 | Yes | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-1\) | \(-1\) |
\(\chi_{44}(41,\cdot)\) | 44.f | 10 | No | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-1\) | \(1\) |
\(\chi_{44}(43,\cdot)\) | 44.c | 2 | Yes | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |