from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4304, base_ring=CyclotomicField(268))
M = H._module
chi = DirichletCharacter(H, M([134,201,223]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,4304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4304\) | |
| Conductor: | \(4304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(268\) | 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: | $\Q(\zeta_{268})$ |
| Fixed field: | Number field defined by a degree 268 polynomial (not computed) |
First 31 of 132 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4304}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{221}{268}\right)\) | \(e\left(\frac{217}{268}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{169}{268}\right)\) | \(e\left(\frac{109}{268}\right)\) | \(e\left(\frac{73}{268}\right)\) | \(e\left(\frac{99}{268}\right)\) | \(e\left(\frac{41}{134}\right)\) | \(e\left(\frac{69}{268}\right)\) |
| \(\chi_{4304}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{249}{268}\right)\) | \(e\left(\frac{25}{268}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{17}{268}\right)\) | \(e\left(\frac{141}{268}\right)\) | \(e\left(\frac{1}{268}\right)\) | \(e\left(\frac{251}{268}\right)\) | \(e\left(\frac{85}{134}\right)\) | \(e\left(\frac{45}{268}\right)\) |
| \(\chi_{4304}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{67}\right)\) | \(e\left(\frac{191}{268}\right)\) | \(e\left(\frac{59}{268}\right)\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{83}{268}\right)\) | \(e\left(\frac{247}{268}\right)\) | \(e\left(\frac{131}{268}\right)\) | \(e\left(\frac{185}{268}\right)\) | \(e\left(\frac{13}{134}\right)\) | \(e\left(\frac{267}{268}\right)\) |
| \(\chi_{4304}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{37}{268}\right)\) | \(e\left(\frac{177}{268}\right)\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{249}{268}\right)\) | \(e\left(\frac{205}{268}\right)\) | \(e\left(\frac{125}{268}\right)\) | \(e\left(\frac{19}{268}\right)\) | \(e\left(\frac{39}{134}\right)\) | \(e\left(\frac{265}{268}\right)\) |
| \(\chi_{4304}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{55}{268}\right)\) | \(e\left(\frac{111}{268}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{247}{268}\right)\) | \(e\left(\frac{15}{268}\right)\) | \(e\left(\frac{251}{268}\right)\) | \(e\left(\frac{21}{268}\right)\) | \(e\left(\frac{29}{134}\right)\) | \(e\left(\frac{39}{268}\right)\) |
| \(\chi_{4304}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{19}{268}\right)\) | \(e\left(\frac{243}{268}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{251}{268}\right)\) | \(e\left(\frac{127}{268}\right)\) | \(e\left(\frac{267}{268}\right)\) | \(e\left(\frac{17}{268}\right)\) | \(e\left(\frac{49}{134}\right)\) | \(e\left(\frac{223}{268}\right)\) |
| \(\chi_{4304}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{179}{268}\right)\) | \(e\left(\frac{103}{268}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{263}{268}\right)\) | \(e\left(\frac{195}{268}\right)\) | \(e\left(\frac{47}{268}\right)\) | \(e\left(\frac{5}{268}\right)\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{239}{268}\right)\) |
| \(\chi_{4304}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{225}{268}\right)\) | \(e\left(\frac{113}{268}\right)\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{109}{268}\right)\) | \(e\left(\frac{37}{268}\right)\) | \(e\left(\frac{101}{268}\right)\) | \(e\left(\frac{159}{268}\right)\) | \(e\left(\frac{9}{134}\right)\) | \(e\left(\frac{257}{268}\right)\) |
| \(\chi_{4304}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{173}{268}\right)\) | \(e\left(\frac{125}{268}\right)\) | \(e\left(\frac{50}{67}\right)\) | \(e\left(\frac{85}{268}\right)\) | \(e\left(\frac{169}{268}\right)\) | \(e\left(\frac{5}{268}\right)\) | \(e\left(\frac{183}{268}\right)\) | \(e\left(\frac{23}{134}\right)\) | \(e\left(\frac{225}{268}\right)\) |
| \(\chi_{4304}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{67}\right)\) | \(e\left(\frac{139}{268}\right)\) | \(e\left(\frac{71}{268}\right)\) | \(e\left(\frac{15}{67}\right)\) | \(e\left(\frac{59}{268}\right)\) | \(e\left(\frac{111}{268}\right)\) | \(e\left(\frac{35}{268}\right)\) | \(e\left(\frac{209}{268}\right)\) | \(e\left(\frac{27}{134}\right)\) | \(e\left(\frac{235}{268}\right)\) |
| \(\chi_{4304}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{265}{268}\right)\) | \(e\left(\frac{145}{268}\right)\) | \(e\left(\frac{58}{67}\right)\) | \(e\left(\frac{45}{268}\right)\) | \(e\left(\frac{121}{268}\right)\) | \(e\left(\frac{113}{268}\right)\) | \(e\left(\frac{223}{268}\right)\) | \(e\left(\frac{91}{134}\right)\) | \(e\left(\frac{261}{268}\right)\) |
| \(\chi_{4304}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{233}{268}\right)\) | \(e\left(\frac{173}{268}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{257}{268}\right)\) | \(e\left(\frac{161}{268}\right)\) | \(e\left(\frac{157}{268}\right)\) | \(e\left(\frac{11}{268}\right)\) | \(e\left(\frac{79}{134}\right)\) | \(e\left(\frac{97}{268}\right)\) |
| \(\chi_{4304}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{67}\right)\) | \(e\left(\frac{59}{268}\right)\) | \(e\left(\frac{7}{268}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{187}{268}\right)\) | \(e\left(\frac{211}{268}\right)\) | \(e\left(\frac{11}{268}\right)\) | \(e\left(\frac{81}{268}\right)\) | \(e\left(\frac{131}{134}\right)\) | \(e\left(\frac{227}{268}\right)\) |
| \(\chi_{4304}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{257}{268}\right)\) | \(e\left(\frac{85}{268}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{165}{268}\right)\) | \(e\left(\frac{265}{268}\right)\) | \(e\left(\frac{57}{268}\right)\) | \(e\left(\frac{103}{268}\right)\) | \(e\left(\frac{21}{134}\right)\) | \(e\left(\frac{153}{268}\right)\) |
| \(\chi_{4304}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{9}{268}\right)\) | \(e\left(\frac{101}{268}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{133}{268}\right)\) | \(e\left(\frac{173}{268}\right)\) | \(e\left(\frac{197}{268}\right)\) | \(e\left(\frac{135}{268}\right)\) | \(e\left(\frac{129}{134}\right)\) | \(e\left(\frac{21}{268}\right)\) |
| \(\chi_{4304}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{167}{268}\right)\) | \(e\left(\frac{147}{268}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{175}{268}\right)\) | \(e\left(\frac{143}{268}\right)\) | \(e\left(\frac{231}{268}\right)\) | \(e\left(\frac{93}{268}\right)\) | \(e\left(\frac{71}{134}\right)\) | \(e\left(\frac{211}{268}\right)\) |
| \(\chi_{4304}(355,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{177}{268}\right)\) | \(e\left(\frac{21}{268}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{25}{268}\right)\) | \(e\left(\frac{97}{268}\right)\) | \(e\left(\frac{33}{268}\right)\) | \(e\left(\frac{243}{268}\right)\) | \(e\left(\frac{125}{134}\right)\) | \(e\left(\frac{145}{268}\right)\) |
| \(\chi_{4304}(363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{67}\right)\) | \(e\left(\frac{231}{268}\right)\) | \(e\left(\frac{91}{268}\right)\) | \(e\left(\frac{23}{67}\right)\) | \(e\left(\frac{19}{268}\right)\) | \(e\left(\frac{63}{268}\right)\) | \(e\left(\frac{143}{268}\right)\) | \(e\left(\frac{249}{268}\right)\) | \(e\left(\frac{95}{134}\right)\) | \(e\left(\frac{3}{268}\right)\) |
| \(\chi_{4304}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{165}{268}\right)\) | \(e\left(\frac{65}{268}\right)\) | \(e\left(\frac{26}{67}\right)\) | \(e\left(\frac{205}{268}\right)\) | \(e\left(\frac{45}{268}\right)\) | \(e\left(\frac{217}{268}\right)\) | \(e\left(\frac{63}{268}\right)\) | \(e\left(\frac{87}{134}\right)\) | \(e\left(\frac{117}{268}\right)\) |
| \(\chi_{4304}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{203}{268}\right)\) | \(e\left(\frac{15}{268}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{171}{268}\right)\) | \(e\left(\frac{31}{268}\right)\) | \(e\left(\frac{215}{268}\right)\) | \(e\left(\frac{97}{268}\right)\) | \(e\left(\frac{51}{134}\right)\) | \(e\left(\frac{27}{268}\right)\) |
| \(\chi_{4304}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{71}{268}\right)\) | \(e\left(\frac{231}{268}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{7}{268}\right)\) | \(e\left(\frac{263}{268}\right)\) | \(e\left(\frac{95}{268}\right)\) | \(e\left(\frac{261}{268}\right)\) | \(e\left(\frac{35}{134}\right)\) | \(e\left(\frac{255}{268}\right)\) |
| \(\chi_{4304}(555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{67}\right)\) | \(e\left(\frac{199}{268}\right)\) | \(e\left(\frac{119}{268}\right)\) | \(e\left(\frac{61}{67}\right)\) | \(e\left(\frac{231}{268}\right)\) | \(e\left(\frac{103}{268}\right)\) | \(e\left(\frac{187}{268}\right)\) | \(e\left(\frac{37}{268}\right)\) | \(e\left(\frac{83}{134}\right)\) | \(e\left(\frac{107}{268}\right)\) |
| \(\chi_{4304}(675,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{67}\right)\) | \(e\left(\frac{109}{268}\right)\) | \(e\left(\frac{181}{268}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{241}{268}\right)\) | \(e\left(\frac{249}{268}\right)\) | \(e\left(\frac{93}{268}\right)\) | \(e\left(\frac{27}{268}\right)\) | \(e\left(\frac{133}{134}\right)\) | \(e\left(\frac{165}{268}\right)\) |
| \(\chi_{4304}(691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{117}{268}\right)\) | \(e\left(\frac{241}{268}\right)\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{121}{268}\right)\) | \(e\left(\frac{105}{268}\right)\) | \(e\left(\frac{149}{268}\right)\) | \(e\left(\frac{147}{268}\right)\) | \(e\left(\frac{69}{134}\right)\) | \(e\left(\frac{5}{268}\right)\) |
| \(\chi_{4304}(747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{67}\right)\) | \(e\left(\frac{223}{268}\right)\) | \(e\left(\frac{31}{268}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{139}{268}\right)\) | \(e\left(\frac{207}{268}\right)\) | \(e\left(\frac{87}{268}\right)\) | \(e\left(\frac{129}{268}\right)\) | \(e\left(\frac{25}{134}\right)\) | \(e\left(\frac{163}{268}\right)\) |
| \(\chi_{4304}(795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{107}{268}\right)\) | \(e\left(\frac{99}{268}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{3}{268}\right)\) | \(e\left(\frac{151}{268}\right)\) | \(e\left(\frac{79}{268}\right)\) | \(e\left(\frac{265}{268}\right)\) | \(e\left(\frac{15}{134}\right)\) | \(e\left(\frac{71}{268}\right)\) |
| \(\chi_{4304}(819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{241}{268}\right)\) | \(e\left(\frac{233}{268}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{137}{268}\right)\) | \(e\left(\frac{17}{268}\right)\) | \(e\left(\frac{213}{268}\right)\) | \(e\left(\frac{131}{268}\right)\) | \(e\left(\frac{15}{134}\right)\) | \(e\left(\frac{205}{268}\right)\) |
| \(\chi_{4304}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{67}\right)\) | \(e\left(\frac{89}{268}\right)\) | \(e\left(\frac{165}{268}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{5}{268}\right)\) | \(e\left(\frac{73}{268}\right)\) | \(e\left(\frac{221}{268}\right)\) | \(e\left(\frac{263}{268}\right)\) | \(e\left(\frac{25}{134}\right)\) | \(e\left(\frac{29}{268}\right)\) |
| \(\chi_{4304}(923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{251}{268}\right)\) | \(e\left(\frac{107}{268}\right)\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{255}{268}\right)\) | \(e\left(\frac{239}{268}\right)\) | \(e\left(\frac{15}{268}\right)\) | \(e\left(\frac{13}{268}\right)\) | \(e\left(\frac{69}{134}\right)\) | \(e\left(\frac{139}{268}\right)\) |
| \(\chi_{4304}(939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{67}\right)\) | \(e\left(\frac{243}{268}\right)\) | \(e\left(\frac{47}{268}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{107}{268}\right)\) | \(e\left(\frac{115}{268}\right)\) | \(e\left(\frac{227}{268}\right)\) | \(e\left(\frac{161}{268}\right)\) | \(e\left(\frac{133}{134}\right)\) | \(e\left(\frac{31}{268}\right)\) |
| \(\chi_{4304}(1059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{67}\right)\) | \(e\left(\frac{65}{268}\right)\) | \(e\left(\frac{253}{268}\right)\) | \(e\left(\frac{61}{67}\right)\) | \(e\left(\frac{97}{268}\right)\) | \(e\left(\frac{237}{268}\right)\) | \(e\left(\frac{53}{268}\right)\) | \(e\left(\frac{171}{268}\right)\) | \(e\left(\frac{83}{134}\right)\) | \(e\left(\frac{241}{268}\right)\) |