from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2850, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([45,9,85]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,2850))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2850\) | |
| Conductor: | \(1425\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(90\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1425.cm | 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 90 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2850}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{2850}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{2850}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{2850}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{2850}(509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{2850}(629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{2850}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{2850}(839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{2850}(869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{2850}(1079,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{2850}(1169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{2850}(1229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{2850}(1409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{2850}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{2850}(1739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{2850}(1769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{2850}(1979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{2850}(2009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{2850}(2219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{2850}(2309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{2850}(2339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{2850}(2369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{2850}(2579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{2850}(2789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) |