from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([81,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,209))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(209\) | |
| Conductor: | \(209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(90\) | 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: | $\Q(\zeta_{45})$ |
| Fixed field: | Number field defined by a degree 90 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{209}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(150,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(156,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) |