from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([171,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,1805))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1805\) | |
| Conductor: | \(1805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(342\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 342 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1805}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{145}{342}\right)\) |
| \(\chi_{1805}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{342}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{319}{342}\right)\) |
| \(\chi_{1805}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{35}{342}\right)\) |
| \(\chi_{1805}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{342}\right)\) | \(e\left(\frac{263}{342}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{187}{342}\right)\) |
| \(\chi_{1805}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{342}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{257}{342}\right)\) |
| \(\chi_{1805}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{149}{342}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{301}{342}\right)\) |
| \(\chi_{1805}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{342}\right)\) | \(e\left(\frac{109}{342}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{71}{342}\right)\) |
| \(\chi_{1805}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{342}\right)\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{331}{342}\right)\) |
| \(\chi_{1805}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{263}{342}\right)\) |
| \(\chi_{1805}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{319}{342}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{167}{342}\right)\) |
| \(\chi_{1805}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{197}{342}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{235}{342}\right)\) |
| \(\chi_{1805}(199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{342}\right)\) | \(e\left(\frac{131}{342}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{283}{342}\right)\) |
| \(\chi_{1805}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{342}\right)\) | \(e\left(\frac{145}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{107}{342}\right)\) |
| \(\chi_{1805}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{241}{342}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{317}{342}\right)\) |
| \(\chi_{1805}(264,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{342}\right)\) | \(e\left(\frac{229}{342}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{77}{342}\right)\) |
| \(\chi_{1805}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{342}\right)\) | \(e\left(\frac{71}{342}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{71}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{109}{342}\right)\) |
| \(\chi_{1805}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{265}{342}\right)\) |
| \(\chi_{1805}(309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{143}{342}\right)\) |
| \(\chi_{1805}(329,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{277}{342}\right)\) |
| \(\chi_{1805}(339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{295}{342}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{29}{342}\right)\) |
| \(\chi_{1805}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{139}{342}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{329}{342}\right)\) |
| \(\chi_{1805}(384,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{342}\right)\) | \(e\left(\frac{287}{342}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{325}{342}\right)\) |
| \(\chi_{1805}(404,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{217}{342}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{179}{342}\right)\) |
| \(\chi_{1805}(424,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{342}\right)\) | \(e\left(\frac{155}{342}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{79}{342}\right)\) |
| \(\chi_{1805}(434,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{342}\right)\) | \(e\left(\frac{7}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{83}{342}\right)\) |
| \(\chi_{1805}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{49}{342}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{239}{342}\right)\) |
| \(\chi_{1805}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{199}{342}\right)\) |
| \(\chi_{1805}(484,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{342}\right)\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{229}{342}\right)\) |
| \(\chi_{1805}(499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{215}{342}\right)\) |
| \(\chi_{1805}(519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{299}{342}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{223}{342}\right)\) |
| \(\chi_{1805}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{61}{342}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{137}{342}\right)\) |