from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,47]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,10000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 125.h | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10000}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) |
| \(\chi_{10000}(449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) |
| \(\chi_{10000}(849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) |
| \(\chi_{10000}(1649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) |
| \(\chi_{10000}(2049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) |
| \(\chi_{10000}(2449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) |
| \(\chi_{10000}(2849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) |
| \(\chi_{10000}(3649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) |
| \(\chi_{10000}(4049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) |
| \(\chi_{10000}(4449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) |
| \(\chi_{10000}(4849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) |
| \(\chi_{10000}(5649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) |
| \(\chi_{10000}(6049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) |
| \(\chi_{10000}(6449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) |
| \(\chi_{10000}(6849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) |
| \(\chi_{10000}(7649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) |
| \(\chi_{10000}(8049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) |
| \(\chi_{10000}(8449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{3}{50}\right)\) |
| \(\chi_{10000}(8849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) |
| \(\chi_{10000}(9649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) |