from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([36,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,475))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(475\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(45\) | 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: | $\Q(\zeta_{45})$ |
| Fixed field: | Number field defined by a degree 45 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{475}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) |
| \(\chi_{475}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) |
| \(\chi_{475}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) |
| \(\chi_{475}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) |
| \(\chi_{475}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) |
| \(\chi_{475}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) |
| \(\chi_{475}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) |
| \(\chi_{475}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) |
| \(\chi_{475}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) |
| \(\chi_{475}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) |
| \(\chi_{475}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) |
| \(\chi_{475}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) |
| \(\chi_{475}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) |
| \(\chi_{475}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) |
| \(\chi_{475}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) |
| \(\chi_{475}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) |
| \(\chi_{475}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) |
| \(\chi_{475}(366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) |
| \(\chi_{475}(386,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) |
| \(\chi_{475}(396,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) |
| \(\chi_{475}(416,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) |
| \(\chi_{475}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) |
| \(\chi_{475}(446,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) |
| \(\chi_{475}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) |