from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([270,286]))
chi.galois_orbit()
[g,chi] = znchar(Mod(14,5239))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5239\) | |
| Conductor: | \(5239\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(195\) | 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: | $\Q(\zeta_{195})$ |
| Fixed field: | Number field defined by a degree 195 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5239}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{31}{195}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{34}{195}\right)\) |
| \(\chi_{5239}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{19}{195}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{41}{195}\right)\) | \(e\left(\frac{137}{195}\right)\) | \(e\left(\frac{89}{195}\right)\) |
| \(\chi_{5239}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{77}{195}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{58}{195}\right)\) | \(e\left(\frac{106}{195}\right)\) |
| \(\chi_{5239}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{112}{195}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{29}{195}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{101}{195}\right)\) |
| \(\chi_{5239}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{2}{195}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{148}{195}\right)\) | \(e\left(\frac{76}{195}\right)\) | \(e\left(\frac{112}{195}\right)\) |
| \(\chi_{5239}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{31}{195}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{62}{195}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{68}{195}\right)\) |
| \(\chi_{5239}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{19}{195}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{127}{195}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{38}{195}\right)\) | \(e\left(\frac{146}{195}\right)\) | \(e\left(\frac{92}{195}\right)\) |
| \(\chi_{5239}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{101}{195}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{109}{195}\right)\) | \(e\left(\frac{58}{195}\right)\) |
| \(\chi_{5239}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{194}{195}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{139}{195}\right)\) |
| \(\chi_{5239}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{163}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{131}{195}\right)\) | \(e\left(\frac{62}{195}\right)\) | \(e\left(\frac{194}{195}\right)\) |
| \(\chi_{5239}(534,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{122}{195}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{56}{195}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{178}{195}\right)\) | \(e\left(\frac{16}{195}\right)\) |
| \(\chi_{5239}(547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{136}{195}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{11}{195}\right)\) |
| \(\chi_{5239}(586,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{77}{195}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{1}{195}\right)\) | \(e\left(\frac{22}{195}\right)\) |
| \(\chi_{5239}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{76}{195}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{152}{195}\right)\) | \(e\left(\frac{194}{195}\right)\) | \(e\left(\frac{173}{195}\right)\) |
| \(\chi_{5239}(638,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{71}{195}\right)\) | \(e\left(\frac{2}{195}\right)\) |
| \(\chi_{5239}(794,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{146}{195}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{83}{195}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{97}{195}\right)\) | \(e\left(\frac{34}{195}\right)\) | \(e\left(\frac{163}{195}\right)\) |
| \(\chi_{5239}(820,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{16}{195}\right)\) | \(e\left(\frac{82}{195}\right)\) | \(e\left(\frac{49}{195}\right)\) |
| \(\chi_{5239}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{131}{195}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{139}{195}\right)\) | \(e\left(\frac{103}{195}\right)\) | \(e\left(\frac{121}{195}\right)\) |
| \(\chi_{5239}(950,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{16}{195}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{116}{195}\right)\) |
| \(\chi_{5239}(989,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{164}{195}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{152}{195}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{133}{195}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{127}{195}\right)\) |
| \(\chi_{5239}(1002,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{83}{195}\right)\) |
| \(\chi_{5239}(1041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{109}{195}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{82}{195}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{107}{195}\right)\) |
| \(\chi_{5239}(1197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{73}{195}\right)\) |
| \(\chi_{5239}(1223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{53}{195}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{106}{195}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{154}{195}\right)\) |
| \(\chi_{5239}(1249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{58}{195}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{14}{195}\right)\) |
| \(\chi_{5239}(1340,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{17}{195}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{11}{195}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{34}{195}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{31}{195}\right)\) |
| \(\chi_{5239}(1392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{32}{195}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{46}{195}\right)\) | \(e\left(\frac{37}{195}\right)\) |
| \(\chi_{5239}(1405,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{73}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{137}{195}\right)\) | \(e\left(\frac{44}{195}\right)\) | \(e\left(\frac{188}{195}\right)\) |
| \(\chi_{5239}(1444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{17}{195}\right)\) |
| \(\chi_{5239}(1600,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{41}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{38}{195}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{82}{195}\right)\) | \(e\left(\frac{79}{195}\right)\) | \(e\left(\frac{178}{195}\right)\) |
| \(\chi_{5239}(1626,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{29}{195}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{1}{195}\right)\) | \(e\left(\frac{127}{195}\right)\) | \(e\left(\frac{64}{195}\right)\) |