from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([18,10]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,209))
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: | \(45\) | 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: | 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{209}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{209}(180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{209}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |