from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4598, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([54,40]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,4598))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4598\) | |
| Conductor: | \(209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(45\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 209.u | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4598}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{45}\right)\) |
| \(\chi_{4598}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{45}\right)\) |
| \(\chi_{4598}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{45}\right)\) |
| \(\chi_{4598}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{16}{45}\right)\) |
| \(\chi_{4598}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{44}{45}\right)\) |
| \(\chi_{4598}(807,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{37}{45}\right)\) |
| \(\chi_{4598}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{45}\right)\) |
| \(\chi_{4598}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{23}{45}\right)\) |
| \(\chi_{4598}(1461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{26}{45}\right)\) |
| \(\chi_{4598}(1479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{14}{45}\right)\) |
| \(\chi_{4598}(1697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{45}\right)\) |
| \(\chi_{4598}(1963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{45}\right)\) |
| \(\chi_{4598}(2259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{45}\right)\) |
| \(\chi_{4598}(2429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{41}{45}\right)\) |
| \(\chi_{4598}(2665,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{38}{45}\right)\) |
| \(\chi_{4598}(2913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{31}{45}\right)\) |
| \(\chi_{4598}(2931,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{29}{45}\right)\) |
| \(\chi_{4598}(3227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{32}{45}\right)\) |
| \(\chi_{4598}(3391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{28}{45}\right)\) |
| \(\chi_{4598}(3633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{43}{45}\right)\) |
| \(\chi_{4598}(3657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{45}\right)\) |
| \(\chi_{4598}(3711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{22}{45}\right)\) |
| \(\chi_{4598}(3881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{45}\right)\) |
| \(\chi_{4598}(3899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{34}{45}\right)\) |