from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([117,100]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,475))
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: | \(180\) | 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: | $\Q(\zeta_{180})$ |
| Fixed field: | Number field defined by a degree 180 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{475}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{180}\right)\) |
| \(\chi_{475}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{180}\right)\) |
| \(\chi_{475}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{157}{180}\right)\) |
| \(\chi_{475}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{163}{180}\right)\) |
| \(\chi_{475}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{67}{180}\right)\) |
| \(\chi_{475}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{179}{180}\right)\) |
| \(\chi_{475}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{169}{180}\right)\) |
| \(\chi_{475}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{133}{180}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{101}{180}\right)\) |
| \(\chi_{475}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{83}{180}\right)\) |
| \(\chi_{475}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{59}{180}\right)\) |
| \(\chi_{475}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{113}{180}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{121}{180}\right)\) |
| \(\chi_{475}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{180}\right)\) | \(e\left(\frac{107}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{19}{180}\right)\) |
| \(\chi_{475}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{180}\right)\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{89}{180}\right)\) |
| \(\chi_{475}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{59}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{103}{180}\right)\) |
| \(\chi_{475}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{180}\right)\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{133}{180}\right)\) |
| \(\chi_{475}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{151}{180}\right)\) |
| \(\chi_{475}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{7}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{119}{180}\right)\) |
| \(\chi_{475}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{180}\right)\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{149}{180}\right)\) |
| \(\chi_{475}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{109}{180}\right)\) |
| \(\chi_{475}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{180}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{53}{180}\right)\) |
| \(\chi_{475}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{167}{180}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{139}{180}\right)\) |
| \(\chi_{475}(252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{163}{180}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{71}{180}\right)\) |
| \(\chi_{475}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{180}\right)\) | \(e\left(\frac{101}{180}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{97}{180}\right)\) |
| \(\chi_{475}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{180}\right)\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{180}\right)\) |
| \(\chi_{475}(272,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{180}\right)\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{180}\right)\) |
| \(\chi_{475}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{49}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{113}{180}\right)\) |
| \(\chi_{475}(302,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{131}{180}\right)\) |
| \(\chi_{475}(308,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{89}{180}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{73}{180}\right)\) |
| \(\chi_{475}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{180}\right)\) | \(e\left(\frac{77}{180}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{49}{180}\right)\) |
| \(\chi_{475}(327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{91}{180}\right)\) |
| \(\chi_{475}(328,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{180}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{17}{180}\right)\) |