from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,500))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(500\) | |
| Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{500}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) |
| \(\chi_{500}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) |
| \(\chi_{500}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) |
| \(\chi_{500}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) |
| \(\chi_{500}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) |
| \(\chi_{500}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) |
| \(\chi_{500}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
| \(\chi_{500}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) |
| \(\chi_{500}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
| \(\chi_{500}(239,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) |
| \(\chi_{500}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) |
| \(\chi_{500}(279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) |
| \(\chi_{500}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) |
| \(\chi_{500}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
| \(\chi_{500}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) |
| \(\chi_{500}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) |
| \(\chi_{500}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
| \(\chi_{500}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) |
| \(\chi_{500}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) |
| \(\chi_{500}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) |