from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(101, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,101))
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: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | 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 50 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{101}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(-1\) | \(e\left(\frac{13}{50}\right)\) |
| \(\chi_{101}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(-1\) | \(e\left(\frac{47}{50}\right)\) |
| \(\chi_{101}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(-1\) | \(e\left(\frac{29}{50}\right)\) |
| \(\chi_{101}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(-1\) | \(e\left(\frac{19}{50}\right)\) |
| \(\chi_{101}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(-1\) | \(e\left(\frac{7}{50}\right)\) |
| \(\chi_{101}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(-1\) | \(e\left(\frac{41}{50}\right)\) |
| \(\chi_{101}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(-1\) | \(e\left(\frac{9}{50}\right)\) |
| \(\chi_{101}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(-1\) | \(e\left(\frac{11}{50}\right)\) |
| \(\chi_{101}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(-1\) | \(e\left(\frac{33}{50}\right)\) |
| \(\chi_{101}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(-1\) | \(e\left(\frac{23}{50}\right)\) |
| \(\chi_{101}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(-1\) | \(e\left(\frac{3}{50}\right)\) |
| \(\chi_{101}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(-1\) | \(e\left(\frac{27}{50}\right)\) |
| \(\chi_{101}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(-1\) | \(e\left(\frac{17}{50}\right)\) |
| \(\chi_{101}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(-1\) | \(e\left(\frac{39}{50}\right)\) |
| \(\chi_{101}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(-1\) | \(e\left(\frac{21}{50}\right)\) |
| \(\chi_{101}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(-1\) | \(e\left(\frac{37}{50}\right)\) |
| \(\chi_{101}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(-1\) | \(e\left(\frac{43}{50}\right)\) |
| \(\chi_{101}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(-1\) | \(e\left(\frac{49}{50}\right)\) |
| \(\chi_{101}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(-1\) | \(e\left(\frac{1}{50}\right)\) |
| \(\chi_{101}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(-1\) | \(e\left(\frac{31}{50}\right)\) |