from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(139, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,139))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(139\) | |
| Conductor: | \(139\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(138\) | 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: | $\Q(\zeta_{69})$ |
| Fixed field: | Number field defined by a degree 138 polynomial (not computed) |
First 31 of 44 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{139}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{138}\right)\) | \(e\left(\frac{41}{138}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{38}{69}\right)\) |
| \(\chi_{139}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{138}\right)\) | \(e\left(\frac{25}{138}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{40}{69}\right)\) |
| \(\chi_{139}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{47}{69}\right)\) |
| \(\chi_{139}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{101}{138}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{65}{69}\right)\) |
| \(\chi_{139}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{109}{138}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{64}{69}\right)\) |
| \(\chi_{139}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{138}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{49}{69}\right)\) |
| \(\chi_{139}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{17}{138}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{41}{69}\right)\) |
| \(\chi_{139}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{5}{138}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{8}{69}\right)\) |
| \(\chi_{139}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{121}{138}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{28}{69}\right)\) |
| \(\chi_{139}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{55}{69}\right)\) |
| \(\chi_{139}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{138}\right)\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{52}{69}\right)\) |
| \(\chi_{139}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{138}\right)\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{1}{69}\right)\) |
| \(\chi_{139}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{138}\right)\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{19}{69}\right)\) |
| \(\chi_{139}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{133}{138}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{61}{69}\right)\) |
| \(\chi_{139}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{103}{138}\right)\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{13}{69}\right)\) |
| \(\chi_{139}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{138}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{22}{69}\right)\) |
| \(\chi_{139}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{138}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{26}{69}\right)\) |
| \(\chi_{139}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{138}\right)\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{2}{69}\right)\) |
| \(\chi_{139}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{97}{138}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{31}{69}\right)\) |
| \(\chi_{139}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{138}\right)\) | \(e\left(\frac{35}{138}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{56}{69}\right)\) |
| \(\chi_{139}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{138}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{68}{69}\right)\) |
| \(\chi_{139}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{20}{69}\right)\) |
| \(\chi_{139}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{138}\right)\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{35}{69}\right)\) |
| \(\chi_{139}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{29}{138}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{5}{69}\right)\) |
| \(\chi_{139}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{138}\right)\) | \(e\left(\frac{85}{138}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{67}{69}\right)\) |
| \(\chi_{139}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{138}\right)\) | \(e\left(\frac{113}{138}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{29}{69}\right)\) |
| \(\chi_{139}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{138}\right)\) | \(e\left(\frac{1}{138}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{43}{69}\right)\) |
| \(\chi_{139}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{138}\right)\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{10}{69}\right)\) |
| \(\chi_{139}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{138}\right)\) | \(e\left(\frac{37}{138}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{4}{69}\right)\) |
| \(\chi_{139}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{62}{69}\right)\) |
| \(\chi_{139}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{19}{138}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{58}{69}\right)\) |