from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([215,104]))
chi.galois_orbit()
[g,chi] = znchar(Mod(82,5239))
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: | \(390\) | 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 390 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5239}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{371}{390}\right)\) | \(e\left(\frac{122}{195}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{119}{130}\right)\) |
| \(\chi_{5239}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{131}{195}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{67}{195}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{107}{130}\right)\) |
| \(\chi_{5239}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{390}\right)\) | \(e\left(\frac{89}{195}\right)\) | \(e\left(\frac{122}{195}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{178}{195}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) |
| \(\chi_{5239}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{390}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{137}{195}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{17}{130}\right)\) |
| \(\chi_{5239}(231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{390}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{86}{195}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{51}{130}\right)\) |
| \(\chi_{5239}(348,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{172}{195}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) |
| \(\chi_{5239}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{390}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{16}{195}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{11}{130}\right)\) |
| \(\chi_{5239}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{4}{195}\right)\) | \(e\left(\frac{172}{195}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{8}{195}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{103}{130}\right)\) |
| \(\chi_{5239}(537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{390}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) |
| \(\chi_{5239}(576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{390}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{103}{195}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) |
| \(\chi_{5239}(608,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{390}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{58}{195}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) |
| \(\chi_{5239}(634,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{390}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{34}{195}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{56}{195}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{71}{130}\right)\) |
| \(\chi_{5239}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{390}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{29}{130}\right)\) |
| \(\chi_{5239}(784,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{390}\right)\) | \(e\left(\frac{68}{195}\right)\) | \(e\left(\frac{194}{195}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{136}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) |
| \(\chi_{5239}(803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{390}\right)\) | \(e\left(\frac{184}{195}\right)\) | \(e\left(\frac{112}{195}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{123}{130}\right)\) |
| \(\chi_{5239}(888,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{390}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{71}{195}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) |
| \(\chi_{5239}(940,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{390}\right)\) | \(e\left(\frac{56}{195}\right)\) | \(e\left(\frac{68}{195}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{112}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) |
| \(\chi_{5239}(979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{390}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{17}{195}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) |
| \(\chi_{5239}(1011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{390}\right)\) | \(e\left(\frac{136}{195}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{77}{195}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) |
| \(\chi_{5239}(1154,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{390}\right)\) | \(e\left(\frac{142}{195}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{89}{195}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{49}{130}\right)\) |
| \(\chi_{5239}(1187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{390}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{44}{195}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) |
| \(\chi_{5239}(1291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{390}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{19}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) |
| \(\chi_{5239}(1343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{390}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) |
| \(\chi_{5239}(1382,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{390}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{62}{195}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{148}{195}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{53}{130}\right)\) |
| \(\chi_{5239}(1414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{390}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{133}{195}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) |
| \(\chi_{5239}(1440,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{390}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{109}{195}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) |
| \(\chi_{5239}(1557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{390}\right)\) | \(e\left(\frac{127}{195}\right)\) | \(e\left(\frac{1}{195}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{59}{195}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) |
| \(\chi_{5239}(1590,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{390}\right)\) | \(e\left(\frac{188}{195}\right)\) | \(e\left(\frac{89}{195}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) |
| \(\chi_{5239}(1609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) |
| \(\chi_{5239}(1694,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{390}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{161}{195}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{139}{195}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) |
| \(\chi_{5239}(1746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{390}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{157}{195}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{47}{130}\right)\) |