from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9126, base_ring=CyclotomicField(234))
M = H._module
chi = DirichletCharacter(H, M([130,27]))
chi.galois_orbit()
[g,chi] = znchar(Mod(25,9126))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9126\) | |
| Conductor: | \(4563\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(234\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4563.dh | 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: | $\Q(\zeta_{117})$ |
| Fixed field: | Number field defined by a degree 234 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9126}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{25}{234}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{20}{117}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{2}{39}\right)\) |
| \(\chi_{9126}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{234}\right)\) | \(e\left(\frac{77}{234}\right)\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{10}{117}\right)\) | \(e\left(\frac{28}{117}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{9126}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{234}\right)\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{116}{117}\right)\) | \(e\left(\frac{44}{117}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{20}{39}\right)\) |
| \(\chi_{9126}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{187}{234}\right)\) | \(e\left(\frac{85}{234}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{9126}(571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{95}{234}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{94}{117}\right)\) | \(e\left(\frac{76}{117}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{9126}(727,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{115}{234}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{107}{234}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{9126}(805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{117}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{10}{39}\right)\) |
| \(\chi_{9126}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{85}{234}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{117}\right)\) | \(e\left(\frac{116}{117}\right)\) | \(e\left(\frac{23}{234}\right)\) | \(e\left(\frac{35}{39}\right)\) |
| \(\chi_{9126}(1039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{107}{234}\right)\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{61}{117}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{28}{39}\right)\) |
| \(\chi_{9126}(1195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{234}\right)\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{23}{117}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{14}{39}\right)\) |
| \(\chi_{9126}(1273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{103}{117}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{9126}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{217}{234}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{47}{117}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{9126}(1507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{215}{234}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{28}{117}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{139}{234}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{9126}(1663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{234}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{1}{234}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{117}\right)\) | \(e\left(\frac{71}{117}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{11}{39}\right)\) |
| \(\chi_{9126}(1741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{234}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{11}{234}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{70}{117}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{9126}(1897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{115}{234}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{59}{117}\right)\) | \(e\left(\frac{95}{117}\right)\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{29}{39}\right)\) |
| \(\chi_{9126}(1975,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{234}\right)\) | \(e\left(\frac{137}{234}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{112}{117}\right)\) | \(e\left(\frac{103}{117}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{9126}(2131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{101}{117}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{9126}(2209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{71}{234}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{37}{117}\right)\) | \(e\left(\frac{10}{117}\right)\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{1}{39}\right)\) |
| \(\chi_{9126}(2443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{234}\right)\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{34}{117}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{19}{39}\right)\) |
| \(\chi_{9126}(2599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{79}{234}\right)\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{137}{234}\right)\) | \(e\left(\frac{5}{39}\right)\) |
| \(\chi_{9126}(2677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{234}\right)\) | \(e\left(\frac{101}{234}\right)\) | \(e\left(\frac{131}{234}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{117}\right)\) | \(e\left(\frac{58}{117}\right)\) | \(e\left(\frac{187}{234}\right)\) | \(e\left(\frac{37}{39}\right)\) |
| \(\chi_{9126}(2833,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{234}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{151}{234}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{110}{117}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{23}{39}\right)\) |
| \(\chi_{9126}(2911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{167}{234}\right)\) | \(e\left(\frac{161}{234}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{82}{117}\right)\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{16}{39}\right)\) |
| \(\chi_{9126}(3067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{98}{117}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{2}{39}\right)\) |
| \(\chi_{9126}(3145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{233}{234}\right)\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{88}{117}\right)\) | \(e\left(\frac{106}{117}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{9126}(3301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{234}\right)\) | \(e\left(\frac{43}{234}\right)\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{77}{117}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{20}{39}\right)\) |
| \(\chi_{9126}(3535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{117}\right)\) | \(e\left(\frac{29}{117}\right)\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{9126}(3613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{131}{234}\right)\) | \(e\left(\frac{17}{234}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{37}{117}\right)\) | \(e\left(\frac{85}{234}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{9126}(3769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{234}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{44}{117}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{9126}(3847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{197}{234}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{61}{117}\right)\) | \(e\left(\frac{1}{234}\right)\) | \(e\left(\frac{10}{39}\right)\) |