from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3307, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([119]))
chi.galois_orbit()
[g,chi] = znchar(Mod(72,3307))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3307\) | |
| Conductor: | \(3307\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(174\) | 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: | $\Q(\zeta_{87})$ |
| Fixed field: | Number field defined by a degree 174 polynomial (not computed) |
First 31 of 56 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3307}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{174}\right)\) | \(e\left(\frac{23}{174}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{71}{174}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{52}{87}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{23}{87}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{68}{87}\right)\) |
| \(\chi_{3307}(220,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{174}\right)\) | \(e\left(\frac{107}{174}\right)\) | \(e\left(\frac{77}{87}\right)\) | \(e\left(\frac{5}{174}\right)\) | \(e\left(\frac{5}{87}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{20}{87}\right)\) | \(e\left(\frac{41}{87}\right)\) | \(e\left(\frac{44}{87}\right)\) |
| \(\chi_{3307}(334,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{174}\right)\) | \(e\left(\frac{25}{174}\right)\) | \(e\left(\frac{31}{87}\right)\) | \(e\left(\frac{115}{174}\right)\) | \(e\left(\frac{28}{87}\right)\) | \(e\left(\frac{83}{87}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{25}{87}\right)\) | \(e\left(\frac{73}{87}\right)\) | \(e\left(\frac{55}{87}\right)\) |
| \(\chi_{3307}(377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{174}\right)\) | \(e\left(\frac{55}{174}\right)\) | \(e\left(\frac{16}{87}\right)\) | \(e\left(\frac{79}{174}\right)\) | \(e\left(\frac{79}{87}\right)\) | \(e\left(\frac{26}{87}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{34}{87}\right)\) |
| \(\chi_{3307}(468,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{174}\right)\) | \(e\left(\frac{49}{174}\right)\) | \(e\left(\frac{19}{87}\right)\) | \(e\left(\frac{121}{174}\right)\) | \(e\left(\frac{34}{87}\right)\) | \(e\left(\frac{20}{87}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{70}{87}\right)\) | \(e\left(\frac{73}{87}\right)\) |
| \(\chi_{3307}(500,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{174}\right)\) | \(e\left(\frac{119}{174}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{95}{174}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{61}{87}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{83}{87}\right)\) | \(e\left(\frac{53}{87}\right)\) |
| \(\chi_{3307}(695,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{174}\right)\) | \(e\left(\frac{43}{174}\right)\) | \(e\left(\frac{22}{87}\right)\) | \(e\left(\frac{163}{174}\right)\) | \(e\left(\frac{76}{87}\right)\) | \(e\left(\frac{14}{87}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{43}{87}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{25}{87}\right)\) |
| \(\chi_{3307}(743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{174}\right)\) | \(e\left(\frac{97}{174}\right)\) | \(e\left(\frac{82}{87}\right)\) | \(e\left(\frac{133}{174}\right)\) | \(e\left(\frac{46}{87}\right)\) | \(e\left(\frac{68}{87}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{10}{87}\right)\) | \(e\left(\frac{64}{87}\right)\) | \(e\left(\frac{22}{87}\right)\) |
| \(\chi_{3307}(763,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{174}\right)\) | \(e\left(\frac{61}{174}\right)\) | \(e\left(\frac{13}{87}\right)\) | \(e\left(\frac{37}{174}\right)\) | \(e\left(\frac{37}{87}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{61}{87}\right)\) | \(e\left(\frac{25}{87}\right)\) | \(e\left(\frac{82}{87}\right)\) |
| \(\chi_{3307}(802,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{174}\right)\) | \(e\left(\frac{149}{174}\right)\) | \(e\left(\frac{56}{87}\right)\) | \(e\left(\frac{59}{174}\right)\) | \(e\left(\frac{59}{87}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{14}{87}\right)\) | \(e\left(\frac{32}{87}\right)\) |
| \(\chi_{3307}(882,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{174}\right)\) | \(e\left(\frac{137}{174}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{143}{174}\right)\) | \(e\left(\frac{56}{87}\right)\) | \(e\left(\frac{79}{87}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{50}{87}\right)\) | \(e\left(\frac{59}{87}\right)\) | \(e\left(\frac{23}{87}\right)\) |
| \(\chi_{3307}(984,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{174}\right)\) | \(e\left(\frac{65}{174}\right)\) | \(e\left(\frac{11}{87}\right)\) | \(e\left(\frac{125}{174}\right)\) | \(e\left(\frac{38}{87}\right)\) | \(e\left(\frac{7}{87}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{68}{87}\right)\) | \(e\left(\frac{56}{87}\right)\) |
| \(\chi_{3307}(1073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{174}\right)\) | \(e\left(\frac{157}{174}\right)\) | \(e\left(\frac{52}{87}\right)\) | \(e\left(\frac{61}{174}\right)\) | \(e\left(\frac{61}{87}\right)\) | \(e\left(\frac{41}{87}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{70}{87}\right)\) | \(e\left(\frac{13}{87}\right)\) | \(e\left(\frac{67}{87}\right)\) |
| \(\chi_{3307}(1205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{174}\right)\) | \(e\left(\frac{127}{174}\right)\) | \(e\left(\frac{67}{87}\right)\) | \(e\left(\frac{97}{174}\right)\) | \(e\left(\frac{10}{87}\right)\) | \(e\left(\frac{11}{87}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{40}{87}\right)\) | \(e\left(\frac{82}{87}\right)\) | \(e\left(\frac{1}{87}\right)\) |
| \(\chi_{3307}(1332,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{174}\right)\) | \(e\left(\frac{151}{174}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{103}{174}\right)\) | \(e\left(\frac{16}{87}\right)\) | \(e\left(\frac{35}{87}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{64}{87}\right)\) | \(e\left(\frac{79}{87}\right)\) | \(e\left(\frac{19}{87}\right)\) |
| \(\chi_{3307}(1388,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{174}\right)\) | \(e\left(\frac{109}{174}\right)\) | \(e\left(\frac{76}{87}\right)\) | \(e\left(\frac{49}{174}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{80}{87}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{22}{87}\right)\) | \(e\left(\frac{19}{87}\right)\) | \(e\left(\frac{31}{87}\right)\) |
| \(\chi_{3307}(1430,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{174}\right)\) | \(e\left(\frac{133}{174}\right)\) | \(e\left(\frac{64}{87}\right)\) | \(e\left(\frac{55}{174}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{17}{87}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{46}{87}\right)\) | \(e\left(\frac{16}{87}\right)\) | \(e\left(\frac{49}{87}\right)\) |
| \(\chi_{3307}(1437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{174}\right)\) | \(e\left(\frac{131}{174}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{11}{174}\right)\) | \(e\left(\frac{11}{87}\right)\) | \(e\left(\frac{73}{87}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{44}{87}\right)\) | \(e\left(\frac{38}{87}\right)\) | \(e\left(\frac{62}{87}\right)\) |
| \(\chi_{3307}(1557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{174}\right)\) | \(e\left(\frac{89}{174}\right)\) | \(e\left(\frac{86}{87}\right)\) | \(e\left(\frac{131}{174}\right)\) | \(e\left(\frac{44}{87}\right)\) | \(e\left(\frac{31}{87}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{2}{87}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{74}{87}\right)\) |
| \(\chi_{3307}(1609,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{174}\right)\) | \(e\left(\frac{103}{174}\right)\) | \(e\left(\frac{79}{87}\right)\) | \(e\left(\frac{91}{174}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{74}{87}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{16}{87}\right)\) | \(e\left(\frac{85}{87}\right)\) | \(e\left(\frac{70}{87}\right)\) |
| \(\chi_{3307}(1635,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{174}\right)\) | \(e\left(\frac{41}{174}\right)\) | \(e\left(\frac{23}{87}\right)\) | \(e\left(\frac{119}{174}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{70}{87}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{41}{87}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{38}{87}\right)\) |
| \(\chi_{3307}(1647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{174}\right)\) | \(e\left(\frac{113}{174}\right)\) | \(e\left(\frac{74}{87}\right)\) | \(e\left(\frac{137}{174}\right)\) | \(e\left(\frac{50}{87}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{26}{87}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{5}{87}\right)\) |
| \(\chi_{3307}(1661,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{174}\right)\) | \(e\left(\frac{37}{174}\right)\) | \(e\left(\frac{25}{87}\right)\) | \(e\left(\frac{31}{174}\right)\) | \(e\left(\frac{31}{87}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{37}{87}\right)\) | \(e\left(\frac{28}{87}\right)\) | \(e\left(\frac{64}{87}\right)\) |
| \(\chi_{3307}(1669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{174}\right)\) | \(e\left(\frac{19}{174}\right)\) | \(e\left(\frac{34}{87}\right)\) | \(e\left(\frac{157}{174}\right)\) | \(e\left(\frac{70}{87}\right)\) | \(e\left(\frac{77}{87}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{19}{87}\right)\) | \(e\left(\frac{52}{87}\right)\) | \(e\left(\frac{7}{87}\right)\) |
| \(\chi_{3307}(1756,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{174}\right)\) | \(e\left(\frac{79}{174}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{85}{174}\right)\) | \(e\left(\frac{85}{87}\right)\) | \(e\left(\frac{50}{87}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{79}{87}\right)\) | \(e\left(\frac{1}{87}\right)\) | \(e\left(\frac{52}{87}\right)\) |
| \(\chi_{3307}(1782,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{174}\right)\) | \(e\left(\frac{85}{174}\right)\) | \(e\left(\frac{1}{87}\right)\) | \(e\left(\frac{43}{174}\right)\) | \(e\left(\frac{43}{87}\right)\) | \(e\left(\frac{56}{87}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{85}{87}\right)\) | \(e\left(\frac{22}{87}\right)\) | \(e\left(\frac{13}{87}\right)\) |
| \(\chi_{3307}(1853,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{174}\right)\) | \(e\left(\frac{115}{174}\right)\) | \(e\left(\frac{73}{87}\right)\) | \(e\left(\frac{7}{174}\right)\) | \(e\left(\frac{7}{87}\right)\) | \(e\left(\frac{86}{87}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{28}{87}\right)\) | \(e\left(\frac{40}{87}\right)\) | \(e\left(\frac{79}{87}\right)\) |
| \(\chi_{3307}(1895,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{174}\right)\) | \(e\left(\frac{71}{174}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{83}{174}\right)\) | \(e\left(\frac{83}{87}\right)\) | \(e\left(\frac{13}{87}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{2}{87}\right)\) | \(e\left(\frac{17}{87}\right)\) |
| \(\chi_{3307}(1906,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{174}\right)\) | \(e\left(\frac{1}{174}\right)\) | \(e\left(\frac{43}{87}\right)\) | \(e\left(\frac{109}{174}\right)\) | \(e\left(\frac{22}{87}\right)\) | \(e\left(\frac{59}{87}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{1}{87}\right)\) | \(e\left(\frac{76}{87}\right)\) | \(e\left(\frac{37}{87}\right)\) |
| \(\chi_{3307}(1952,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{174}\right)\) | \(e\left(\frac{121}{174}\right)\) | \(e\left(\frac{70}{87}\right)\) | \(e\left(\frac{139}{174}\right)\) | \(e\left(\frac{52}{87}\right)\) | \(e\left(\frac{5}{87}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{34}{87}\right)\) | \(e\left(\frac{61}{87}\right)\) | \(e\left(\frac{40}{87}\right)\) |
| \(\chi_{3307}(2081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{174}\right)\) | \(e\left(\frac{95}{174}\right)\) | \(e\left(\frac{83}{87}\right)\) | \(e\left(\frac{89}{174}\right)\) | \(e\left(\frac{2}{87}\right)\) | \(e\left(\frac{37}{87}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{86}{87}\right)\) | \(e\left(\frac{35}{87}\right)\) |