from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,169))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(169\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(156\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{156})$ |
| Fixed field: | Number field defined by a degree 156 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{169}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{103}{156}\right)\) |
| \(\chi_{169}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{83}{156}\right)\) |
| \(\chi_{169}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{101}{156}\right)\) |
| \(\chi_{169}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{1}{156}\right)\) |
| \(\chi_{169}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{127}{156}\right)\) |
| \(\chi_{169}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{41}{156}\right)\) |
| \(\chi_{169}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{133}{156}\right)\) |
| \(\chi_{169}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{151}{156}\right)\) |
| \(\chi_{169}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{47}{156}\right)\) |
| \(\chi_{169}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{137}{156}\right)\) |
| \(\chi_{169}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{109}{156}\right)\) |
| \(\chi_{169}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{19}{156}\right)\) |
| \(\chi_{169}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{107}{156}\right)\) |
| \(\chi_{169}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{77}{156}\right)\) |
| \(\chi_{169}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{85}{156}\right)\) |
| \(\chi_{169}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{43}{156}\right)\) |
| \(\chi_{169}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{11}{156}\right)\) |
| \(\chi_{169}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{17}{156}\right)\) |
| \(\chi_{169}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{61}{156}\right)\) |
| \(\chi_{169}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{67}{156}\right)\) |
| \(\chi_{169}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{71}{156}\right)\) |
| \(\chi_{169}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{113}{156}\right)\) |
| \(\chi_{169}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{37}{156}\right)\) |
| \(\chi_{169}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{131}{156}\right)\) |
| \(\chi_{169}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{53}{156}\right)\) |
| \(\chi_{169}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{115}{156}\right)\) |
| \(\chi_{169}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{35}{156}\right)\) |
| \(\chi_{169}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{149}{156}\right)\) |
| \(\chi_{169}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{145}{156}\right)\) |
| \(\chi_{169}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{139}{156}\right)\) |
| \(\chi_{169}(110,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{95}{156}\right)\) |