from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([54,40]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,407))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(407\) | |
| Conductor: | \(407\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{407}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |