from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([50,42]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,539))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(539\) | |
| Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(105\) | 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: | $\Q(\zeta_{105})$ |
| Fixed field: | Number field defined by a degree 105 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{539}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{539}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{6}{35}\right)\) |
| \(\chi_{539}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{539}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{539}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{539}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{539}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{539}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{539}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{539}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{26}{35}\right)\) |
| \(\chi_{539}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{539}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{23}{35}\right)\) |
| \(\chi_{539}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{539}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{26}{35}\right)\) |
| \(\chi_{539}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{539}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{539}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{539}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{539}(170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{9}{35}\right)\) |
| \(\chi_{539}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{539}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{539}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{539}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{539}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{539}(240,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{539}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{539}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{35}\right)\) |
| \(\chi_{539}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{539}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{539}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{35}\right)\) |
| \(\chi_{539}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{24}{35}\right)\) |