from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(101, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([12]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,101))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(101\) | |
| Conductor: | \(101\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | 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 25 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{101}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(1\) | \(e\left(\frac{3}{25}\right)\) |
| \(\chi_{101}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(1\) | \(e\left(\frac{13}{25}\right)\) |
| \(\chi_{101}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(1\) | \(e\left(\frac{12}{25}\right)\) |
| \(\chi_{101}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(1\) | \(e\left(\frac{9}{25}\right)\) |
| \(\chi_{101}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(1\) | \(e\left(\frac{6}{25}\right)\) |
| \(\chi_{101}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(1\) | \(e\left(\frac{23}{25}\right)\) |
| \(\chi_{101}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(1\) | \(e\left(\frac{7}{25}\right)\) |
| \(\chi_{101}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(1\) | \(e\left(\frac{21}{25}\right)\) |
| \(\chi_{101}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(1\) | \(e\left(\frac{1}{25}\right)\) |
| \(\chi_{101}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(1\) | \(e\left(\frac{14}{25}\right)\) |
| \(\chi_{101}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(1\) | \(e\left(\frac{24}{25}\right)\) |
| \(\chi_{101}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(1\) | \(e\left(\frac{4}{25}\right)\) |
| \(\chi_{101}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(1\) | \(e\left(\frac{18}{25}\right)\) |
| \(\chi_{101}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(1\) | \(e\left(\frac{17}{25}\right)\) |
| \(\chi_{101}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(1\) | \(e\left(\frac{8}{25}\right)\) |
| \(\chi_{101}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(1\) | \(e\left(\frac{16}{25}\right)\) |
| \(\chi_{101}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(1\) | \(e\left(\frac{22}{25}\right)\) |
| \(\chi_{101}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(1\) | \(e\left(\frac{2}{25}\right)\) |
| \(\chi_{101}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(1\) | \(e\left(\frac{11}{25}\right)\) |
| \(\chi_{101}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(1\) | \(e\left(\frac{19}{25}\right)\) |