from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1069, base_ring=CyclotomicField(1068))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,1069))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1069\) | |
| Conductor: | \(1069\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1068\) | 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: | $\Q(\zeta_{1068})$ |
| Fixed field: | Number field defined by a degree 1068 polynomial (not computed) |
First 31 of 352 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1069}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{356}\right)\) | \(e\left(\frac{101}{534}\right)\) | \(e\left(\frac{111}{178}\right)\) | \(e\left(\frac{64}{267}\right)\) | \(e\left(\frac{1}{1068}\right)\) | \(e\left(\frac{707}{1068}\right)\) | \(e\left(\frac{155}{356}\right)\) | \(e\left(\frac{101}{267}\right)\) | \(e\left(\frac{55}{1068}\right)\) | \(e\left(\frac{259}{1068}\right)\) |
| \(\chi_{1069}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{356}\right)\) | \(e\left(\frac{385}{534}\right)\) | \(e\left(\frac{157}{178}\right)\) | \(e\left(\frac{125}{267}\right)\) | \(e\left(\frac{707}{1068}\right)\) | \(e\left(\frac{25}{1068}\right)\) | \(e\left(\frac{293}{356}\right)\) | \(e\left(\frac{118}{267}\right)\) | \(e\left(\frac{437}{1068}\right)\) | \(e\left(\frac{485}{1068}\right)\) |
| \(\chi_{1069}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{356}\right)\) | \(e\left(\frac{215}{534}\right)\) | \(e\left(\frac{53}{178}\right)\) | \(e\left(\frac{49}{267}\right)\) | \(e\left(\frac{55}{1068}\right)\) | \(e\left(\frac{437}{1068}\right)\) | \(e\left(\frac{337}{356}\right)\) | \(e\left(\frac{215}{267}\right)\) | \(e\left(\frac{889}{1068}\right)\) | \(e\left(\frac{361}{1068}\right)\) |
| \(\chi_{1069}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{356}\right)\) | \(e\left(\frac{527}{534}\right)\) | \(e\left(\frac{91}{178}\right)\) | \(e\left(\frac{22}{267}\right)\) | \(e\left(\frac{259}{1068}\right)\) | \(e\left(\frac{485}{1068}\right)\) | \(e\left(\frac{273}{356}\right)\) | \(e\left(\frac{260}{267}\right)\) | \(e\left(\frac{361}{1068}\right)\) | \(e\left(\frac{865}{1068}\right)\) |
| \(\chi_{1069}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{356}\right)\) | \(e\left(\frac{211}{534}\right)\) | \(e\left(\frac{105}{178}\right)\) | \(e\left(\frac{176}{267}\right)\) | \(e\left(\frac{203}{1068}\right)\) | \(e\left(\frac{409}{1068}\right)\) | \(e\left(\frac{137}{356}\right)\) | \(e\left(\frac{211}{267}\right)\) | \(e\left(\frac{485}{1068}\right)\) | \(e\left(\frac{245}{1068}\right)\) |
| \(\chi_{1069}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{356}\right)\) | \(e\left(\frac{1}{534}\right)\) | \(e\left(\frac{165}{178}\right)\) | \(e\left(\frac{35}{267}\right)\) | \(e\left(\frac{497}{1068}\right)\) | \(e\left(\frac{7}{1068}\right)\) | \(e\left(\frac{139}{356}\right)\) | \(e\left(\frac{1}{267}\right)\) | \(e\left(\frac{635}{1068}\right)\) | \(e\left(\frac{563}{1068}\right)\) |
| \(\chi_{1069}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{356}\right)\) | \(e\left(\frac{83}{534}\right)\) | \(e\left(\frac{167}{178}\right)\) | \(e\left(\frac{235}{267}\right)\) | \(e\left(\frac{667}{1068}\right)\) | \(e\left(\frac{581}{1068}\right)\) | \(e\left(\frac{145}{356}\right)\) | \(e\left(\frac{83}{267}\right)\) | \(e\left(\frac{373}{1068}\right)\) | \(e\left(\frac{805}{1068}\right)\) |
| \(\chi_{1069}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{356}\right)\) | \(e\left(\frac{49}{534}\right)\) | \(e\left(\frac{75}{178}\right)\) | \(e\left(\frac{113}{267}\right)\) | \(e\left(\frac{857}{1068}\right)\) | \(e\left(\frac{343}{1068}\right)\) | \(e\left(\frac{47}{356}\right)\) | \(e\left(\frac{49}{267}\right)\) | \(e\left(\frac{143}{1068}\right)\) | \(e\left(\frac{887}{1068}\right)\) |
| \(\chi_{1069}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{356}\right)\) | \(e\left(\frac{367}{534}\right)\) | \(e\left(\frac{35}{178}\right)\) | \(e\left(\frac{29}{267}\right)\) | \(e\left(\frac{305}{1068}\right)\) | \(e\left(\frac{967}{1068}\right)\) | \(e\left(\frac{283}{356}\right)\) | \(e\left(\frac{100}{267}\right)\) | \(e\left(\frac{755}{1068}\right)\) | \(e\left(\frac{1031}{1068}\right)\) |
| \(\chi_{1069}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{225}{356}\right)\) | \(e\left(\frac{325}{534}\right)\) | \(e\left(\frac{47}{178}\right)\) | \(e\left(\frac{161}{267}\right)\) | \(e\left(\frac{257}{1068}\right)\) | \(e\left(\frac{139}{1068}\right)\) | \(e\left(\frac{319}{356}\right)\) | \(e\left(\frac{58}{267}\right)\) | \(e\left(\frac{251}{1068}\right)\) | \(e\left(\frac{347}{1068}\right)\) |
| \(\chi_{1069}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{356}\right)\) | \(e\left(\frac{103}{534}\right)\) | \(e\left(\frac{85}{178}\right)\) | \(e\left(\frac{134}{267}\right)\) | \(e\left(\frac{461}{1068}\right)\) | \(e\left(\frac{187}{1068}\right)\) | \(e\left(\frac{255}{356}\right)\) | \(e\left(\frac{103}{267}\right)\) | \(e\left(\frac{791}{1068}\right)\) | \(e\left(\frac{851}{1068}\right)\) |
| \(\chi_{1069}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{356}\right)\) | \(e\left(\frac{391}{534}\right)\) | \(e\left(\frac{79}{178}\right)\) | \(e\left(\frac{68}{267}\right)\) | \(e\left(\frac{485}{1068}\right)\) | \(e\left(\frac{67}{1068}\right)\) | \(e\left(\frac{59}{356}\right)\) | \(e\left(\frac{124}{267}\right)\) | \(e\left(\frac{1043}{1068}\right)\) | \(e\left(\frac{659}{1068}\right)\) |
| \(\chi_{1069}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{356}\right)\) | \(e\left(\frac{197}{534}\right)\) | \(e\left(\frac{109}{178}\right)\) | \(e\left(\frac{220}{267}\right)\) | \(e\left(\frac{721}{1068}\right)\) | \(e\left(\frac{311}{1068}\right)\) | \(e\left(\frac{327}{356}\right)\) | \(e\left(\frac{197}{267}\right)\) | \(e\left(\frac{139}{1068}\right)\) | \(e\left(\frac{907}{1068}\right)\) |
| \(\chi_{1069}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{356}\right)\) | \(e\left(\frac{61}{534}\right)\) | \(e\left(\frac{97}{178}\right)\) | \(e\left(\frac{266}{267}\right)\) | \(e\left(\frac{947}{1068}\right)\) | \(e\left(\frac{961}{1068}\right)\) | \(e\left(\frac{113}{356}\right)\) | \(e\left(\frac{61}{267}\right)\) | \(e\left(\frac{821}{1068}\right)\) | \(e\left(\frac{701}{1068}\right)\) |
| \(\chi_{1069}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{356}\right)\) | \(e\left(\frac{187}{534}\right)\) | \(e\left(\frac{61}{178}\right)\) | \(e\left(\frac{137}{267}\right)\) | \(e\left(\frac{557}{1068}\right)\) | \(e\left(\frac{775}{1068}\right)\) | \(e\left(\frac{183}{356}\right)\) | \(e\left(\frac{187}{267}\right)\) | \(e\left(\frac{731}{1068}\right)\) | \(e\left(\frac{83}{1068}\right)\) |
| \(\chi_{1069}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{356}\right)\) | \(e\left(\frac{509}{534}\right)\) | \(e\left(\frac{147}{178}\right)\) | \(e\left(\frac{193}{267}\right)\) | \(e\left(\frac{925}{1068}\right)\) | \(e\left(\frac{359}{1068}\right)\) | \(e\left(\frac{263}{356}\right)\) | \(e\left(\frac{242}{267}\right)\) | \(e\left(\frac{679}{1068}\right)\) | \(e\left(\frac{343}{1068}\right)\) |
| \(\chi_{1069}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{356}\right)\) | \(e\left(\frac{91}{534}\right)\) | \(e\left(\frac{63}{178}\right)\) | \(e\left(\frac{248}{267}\right)\) | \(e\left(\frac{371}{1068}\right)\) | \(e\left(\frac{637}{1068}\right)\) | \(e\left(\frac{189}{356}\right)\) | \(e\left(\frac{91}{267}\right)\) | \(e\left(\frac{113}{1068}\right)\) | \(e\left(\frac{1037}{1068}\right)\) |
| \(\chi_{1069}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{356}\right)\) | \(e\left(\frac{439}{534}\right)\) | \(e\left(\frac{167}{178}\right)\) | \(e\left(\frac{146}{267}\right)\) | \(e\left(\frac{311}{1068}\right)\) | \(e\left(\frac{937}{1068}\right)\) | \(e\left(\frac{145}{356}\right)\) | \(e\left(\frac{172}{267}\right)\) | \(e\left(\frac{17}{1068}\right)\) | \(e\left(\frac{449}{1068}\right)\) |
| \(\chi_{1069}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{356}\right)\) | \(e\left(\frac{217}{534}\right)\) | \(e\left(\frac{27}{178}\right)\) | \(e\left(\frac{119}{267}\right)\) | \(e\left(\frac{515}{1068}\right)\) | \(e\left(\frac{985}{1068}\right)\) | \(e\left(\frac{81}{356}\right)\) | \(e\left(\frac{217}{267}\right)\) | \(e\left(\frac{557}{1068}\right)\) | \(e\left(\frac{953}{1068}\right)\) |
| \(\chi_{1069}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{356}\right)\) | \(e\left(\frac{35}{534}\right)\) | \(e\left(\frac{79}{178}\right)\) | \(e\left(\frac{157}{267}\right)\) | \(e\left(\frac{841}{1068}\right)\) | \(e\left(\frac{779}{1068}\right)\) | \(e\left(\frac{59}{356}\right)\) | \(e\left(\frac{35}{267}\right)\) | \(e\left(\frac{331}{1068}\right)\) | \(e\left(\frac{1015}{1068}\right)\) |
| \(\chi_{1069}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{356}\right)\) | \(e\left(\frac{71}{534}\right)\) | \(e\left(\frac{145}{178}\right)\) | \(e\left(\frac{82}{267}\right)\) | \(e\left(\frac{43}{1068}\right)\) | \(e\left(\frac{497}{1068}\right)\) | \(e\left(\frac{257}{356}\right)\) | \(e\left(\frac{71}{267}\right)\) | \(e\left(\frac{229}{1068}\right)\) | \(e\left(\frac{457}{1068}\right)\) |
| \(\chi_{1069}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{356}\right)\) | \(e\left(\frac{193}{534}\right)\) | \(e\left(\frac{161}{178}\right)\) | \(e\left(\frac{80}{267}\right)\) | \(e\left(\frac{869}{1068}\right)\) | \(e\left(\frac{283}{1068}\right)\) | \(e\left(\frac{127}{356}\right)\) | \(e\left(\frac{193}{267}\right)\) | \(e\left(\frac{803}{1068}\right)\) | \(e\left(\frac{791}{1068}\right)\) |
| \(\chi_{1069}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{273}{356}\right)\) | \(e\left(\frac{157}{534}\right)\) | \(e\left(\frac{95}{178}\right)\) | \(e\left(\frac{155}{267}\right)\) | \(e\left(\frac{65}{1068}\right)\) | \(e\left(\frac{31}{1068}\right)\) | \(e\left(\frac{107}{356}\right)\) | \(e\left(\frac{157}{267}\right)\) | \(e\left(\frac{371}{1068}\right)\) | \(e\left(\frac{815}{1068}\right)\) |
| \(\chi_{1069}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{356}\right)\) | \(e\left(\frac{443}{534}\right)\) | \(e\left(\frac{115}{178}\right)\) | \(e\left(\frac{19}{267}\right)\) | \(e\left(\frac{697}{1068}\right)\) | \(e\left(\frac{431}{1068}\right)\) | \(e\left(\frac{167}{356}\right)\) | \(e\left(\frac{176}{267}\right)\) | \(e\left(\frac{955}{1068}\right)\) | \(e\left(\frac{31}{1068}\right)\) |
| \(\chi_{1069}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{356}\right)\) | \(e\left(\frac{517}{534}\right)\) | \(e\left(\frac{43}{178}\right)\) | \(e\left(\frac{206}{267}\right)\) | \(e\left(\frac{95}{1068}\right)\) | \(e\left(\frac{949}{1068}\right)\) | \(e\left(\frac{129}{356}\right)\) | \(e\left(\frac{250}{267}\right)\) | \(e\left(\frac{953}{1068}\right)\) | \(e\left(\frac{41}{1068}\right)\) |
| \(\chi_{1069}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{356}\right)\) | \(e\left(\frac{533}{534}\right)\) | \(e\left(\frac{13}{178}\right)\) | \(e\left(\frac{232}{267}\right)\) | \(e\left(\frac{571}{1068}\right)\) | \(e\left(\frac{1061}{1068}\right)\) | \(e\left(\frac{217}{356}\right)\) | \(e\left(\frac{266}{267}\right)\) | \(e\left(\frac{433}{1068}\right)\) | \(e\left(\frac{505}{1068}\right)\) |
| \(\chi_{1069}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{356}\right)\) | \(e\left(\frac{65}{534}\right)\) | \(e\left(\frac{45}{178}\right)\) | \(e\left(\frac{139}{267}\right)\) | \(e\left(\frac{265}{1068}\right)\) | \(e\left(\frac{455}{1068}\right)\) | \(e\left(\frac{135}{356}\right)\) | \(e\left(\frac{65}{267}\right)\) | \(e\left(\frac{691}{1068}\right)\) | \(e\left(\frac{283}{1068}\right)\) |
| \(\chi_{1069}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{356}\right)\) | \(e\left(\frac{227}{534}\right)\) | \(e\left(\frac{75}{178}\right)\) | \(e\left(\frac{202}{267}\right)\) | \(e\left(\frac{145}{1068}\right)\) | \(e\left(\frac{1055}{1068}\right)\) | \(e\left(\frac{47}{356}\right)\) | \(e\left(\frac{227}{267}\right)\) | \(e\left(\frac{499}{1068}\right)\) | \(e\left(\frac{175}{1068}\right)\) |
| \(\chi_{1069}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{356}\right)\) | \(e\left(\frac{493}{534}\right)\) | \(e\left(\frac{177}{178}\right)\) | \(e\left(\frac{167}{267}\right)\) | \(e\left(\frac{449}{1068}\right)\) | \(e\left(\frac{247}{1068}\right)\) | \(e\left(\frac{175}{356}\right)\) | \(e\left(\frac{226}{267}\right)\) | \(e\left(\frac{131}{1068}\right)\) | \(e\left(\frac{947}{1068}\right)\) |
| \(\chi_{1069}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{255}{356}\right)\) | \(e\left(\frac{487}{534}\right)\) | \(e\left(\frac{77}{178}\right)\) | \(e\left(\frac{224}{267}\right)\) | \(e\left(\frac{671}{1068}\right)\) | \(e\left(\frac{205}{1068}\right)\) | \(e\left(\frac{53}{356}\right)\) | \(e\left(\frac{220}{267}\right)\) | \(e\left(\frac{593}{1068}\right)\) | \(e\left(\frac{773}{1068}\right)\) |
| \(\chi_{1069}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{356}\right)\) | \(e\left(\frac{31}{534}\right)\) | \(e\left(\frac{131}{178}\right)\) | \(e\left(\frac{17}{267}\right)\) | \(e\left(\frac{455}{1068}\right)\) | \(e\left(\frac{217}{1068}\right)\) | \(e\left(\frac{37}{356}\right)\) | \(e\left(\frac{31}{267}\right)\) | \(e\left(\frac{461}{1068}\right)\) | \(e\left(\frac{365}{1068}\right)\) |