from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3513, base_ring=CyclotomicField(234))
M = H._module
chi = DirichletCharacter(H, M([0,161]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,3513))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3513\) | |
| Conductor: | \(1171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(234\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1171.u | 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: | $\Q(\zeta_{117})$ |
| Fixed field: | Number field defined by a degree 234 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3513}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{234}\right)\) | \(e\left(\frac{44}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{88}{117}\right)\) |
| \(\chi_{3513}(130,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{110}{117}\right)\) |
| \(\chi_{3513}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{92}{117}\right)\) |
| \(\chi_{3513}(250,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{8}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{229}{234}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{16}{117}\right)\) |
| \(\chi_{3513}(307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{94}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{71}{117}\right)\) |
| \(\chi_{3513}(316,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{234}\right)\) | \(e\left(\frac{11}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{197}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{22}{117}\right)\) |
| \(\chi_{3513}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{89}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{193}{234}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{61}{117}\right)\) |
| \(\chi_{3513}(502,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{73}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{29}{117}\right)\) |
| \(\chi_{3513}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{234}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{41}{117}\right)\) |
| \(\chi_{3513}(580,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{40}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{234}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{80}{117}\right)\) |
| \(\chi_{3513}(598,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{187}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{10}{117}\right)\) |
| \(\chi_{3513}(628,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{85}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{149}{234}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{38}{117}\right)\) |
| \(\chi_{3513}(640,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{4}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{117}\right)\) |
| \(\chi_{3513}(715,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{234}\right)\) | \(e\left(\frac{95}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{43}{234}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{73}{117}\right)\) |
| \(\chi_{3513}(745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{234}\right)\) | \(e\left(\frac{77}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{25}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{37}{117}\right)\) |
| \(\chi_{3513}(781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{55}{117}\right)\) |
| \(\chi_{3513}(850,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{234}\right)\) | \(e\left(\frac{22}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{43}{234}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{44}{117}\right)\) |
| \(\chi_{3513}(904,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{234}\right)\) | \(e\left(\frac{23}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{46}{117}\right)\) |
| \(\chi_{3513}(982,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{234}\right)\) | \(e\left(\frac{20}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{40}{117}\right)\) |
| \(\chi_{3513}(1075,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{234}\right)\) | \(e\left(\frac{32}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{137}{234}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{64}{117}\right)\) |
| \(\chi_{3513}(1084,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{133}{234}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{117}\right)\) |
| \(\chi_{3513}(1090,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{215}{234}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{14}{117}\right)\) |
| \(\chi_{3513}(1162,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{234}\right)\) | \(e\left(\frac{62}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{117}\right)\) |
| \(\chi_{3513}(1174,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{62}{117}\right)\) |
| \(\chi_{3513}(1231,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{34}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{68}{117}\right)\) |
| \(\chi_{3513}(1234,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{106}{117}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{197}{234}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{95}{117}\right)\) |
| \(\chi_{3513}(1318,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{117}\right)\) |
| \(\chi_{3513}(1354,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{234}\right)\) | \(e\left(\frac{116}{117}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{131}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{115}{117}\right)\) |
| \(\chi_{3513}(1414,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{38}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{76}{117}\right)\) |
| \(\chi_{3513}(1429,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{227}{234}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{116}{117}\right)\) |
| \(\chi_{3513}(1447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{101}{117}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{107}{234}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{127}{234}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{85}{117}\right)\) |