from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(858))
M = H._module
chi = DirichletCharacter(H, M([572,642]))
chi.galois_orbit()
[g,chi] = znchar(Mod(74,6013))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6013\) | |
| Conductor: | \(6013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(429\) | 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: | $\Q(\zeta_{429})$ |
| Fixed field: | Number field defined by a degree 429 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6013}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{429}\right)\) | \(e\left(\frac{304}{429}\right)\) | \(e\left(\frac{70}{429}\right)\) | \(e\left(\frac{86}{429}\right)\) | \(e\left(\frac{113}{143}\right)\) | \(e\left(\frac{35}{143}\right)\) | \(e\left(\frac{179}{429}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{103}{429}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{6013}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{274}{429}\right)\) | \(e\left(\frac{2}{429}\right)\) | \(e\left(\frac{119}{429}\right)\) | \(e\left(\frac{232}{429}\right)\) | \(e\left(\frac{92}{143}\right)\) | \(e\left(\frac{131}{143}\right)\) | \(e\left(\frac{4}{429}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{218}{429}\right)\) | \(e\left(\frac{11}{39}\right)\) |
| \(\chi_{6013}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{244}{429}\right)\) | \(e\left(\frac{293}{429}\right)\) | \(e\left(\frac{59}{429}\right)\) | \(e\left(\frac{97}{429}\right)\) | \(e\left(\frac{36}{143}\right)\) | \(e\left(\frac{101}{143}\right)\) | \(e\left(\frac{157}{429}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{191}{429}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{6013}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{429}\right)\) | \(e\left(\frac{356}{429}\right)\) | \(e\left(\frac{161}{429}\right)\) | \(e\left(\frac{112}{429}\right)\) | \(e\left(\frac{74}{143}\right)\) | \(e\left(\frac{9}{143}\right)\) | \(e\left(\frac{283}{429}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{194}{429}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{6013}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{429}\right)\) | \(e\left(\frac{248}{429}\right)\) | \(e\left(\frac{170}{429}\right)\) | \(e\left(\frac{25}{429}\right)\) | \(e\left(\frac{111}{143}\right)\) | \(e\left(\frac{85}{143}\right)\) | \(e\left(\frac{67}{429}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{5}{429}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{6013}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{256}{429}\right)\) | \(e\left(\frac{5}{429}\right)\) | \(e\left(\frac{83}{429}\right)\) | \(e\left(\frac{151}{429}\right)\) | \(e\left(\frac{87}{143}\right)\) | \(e\left(\frac{113}{143}\right)\) | \(e\left(\frac{10}{429}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{116}{429}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{6013}(228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{429}\right)\) | \(e\left(\frac{4}{429}\right)\) | \(e\left(\frac{238}{429}\right)\) | \(e\left(\frac{35}{429}\right)\) | \(e\left(\frac{41}{143}\right)\) | \(e\left(\frac{119}{143}\right)\) | \(e\left(\frac{8}{429}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{7}{429}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{6013}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{380}{429}\right)\) | \(e\left(\frac{175}{429}\right)\) | \(e\left(\frac{331}{429}\right)\) | \(e\left(\frac{137}{429}\right)\) | \(e\left(\frac{42}{143}\right)\) | \(e\left(\frac{94}{143}\right)\) | \(e\left(\frac{350}{429}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{199}{429}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{6013}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{188}{429}\right)\) | \(e\left(\frac{64}{429}\right)\) | \(e\left(\frac{376}{429}\right)\) | \(e\left(\frac{131}{429}\right)\) | \(e\left(\frac{84}{143}\right)\) | \(e\left(\frac{45}{143}\right)\) | \(e\left(\frac{128}{429}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{112}{429}\right)\) | \(e\left(\frac{1}{39}\right)\) |
| \(\chi_{6013}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{429}\right)\) | \(e\left(\frac{226}{429}\right)\) | \(e\left(\frac{148}{429}\right)\) | \(e\left(\frac{47}{429}\right)\) | \(e\left(\frac{100}{143}\right)\) | \(e\left(\frac{74}{143}\right)\) | \(e\left(\frac{23}{429}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{181}{429}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{6013}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{429}\right)\) | \(e\left(\frac{100}{429}\right)\) | \(e\left(\frac{373}{429}\right)\) | \(e\left(\frac{17}{429}\right)\) | \(e\left(\frac{24}{143}\right)\) | \(e\left(\frac{115}{143}\right)\) | \(e\left(\frac{200}{429}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{175}{429}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{6013}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{160}{429}\right)\) | \(e\left(\frac{164}{429}\right)\) | \(e\left(\frac{320}{429}\right)\) | \(e\left(\frac{148}{429}\right)\) | \(e\left(\frac{108}{143}\right)\) | \(e\left(\frac{17}{143}\right)\) | \(e\left(\frac{328}{429}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{287}{429}\right)\) | \(e\left(\frac{5}{39}\right)\) |
| \(\chi_{6013}(340,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{296}{429}\right)\) | \(e\left(\frac{46}{429}\right)\) | \(e\left(\frac{163}{429}\right)\) | \(e\left(\frac{188}{429}\right)\) | \(e\left(\frac{114}{143}\right)\) | \(e\left(\frac{10}{143}\right)\) | \(e\left(\frac{92}{429}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{295}{429}\right)\) | \(e\left(\frac{19}{39}\right)\) |
| \(\chi_{6013}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{362}{429}\right)\) | \(e\left(\frac{178}{429}\right)\) | \(e\left(\frac{295}{429}\right)\) | \(e\left(\frac{56}{429}\right)\) | \(e\left(\frac{37}{143}\right)\) | \(e\left(\frac{76}{143}\right)\) | \(e\left(\frac{356}{429}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{97}{429}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{6013}(354,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{429}\right)\) | \(e\left(\frac{394}{429}\right)\) | \(e\left(\frac{277}{429}\right)\) | \(e\left(\frac{230}{429}\right)\) | \(e\left(\frac{106}{143}\right)\) | \(e\left(\frac{67}{143}\right)\) | \(e\left(\frac{359}{429}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{46}{429}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{6013}(366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{424}{429}\right)\) | \(e\left(\frac{263}{429}\right)\) | \(e\left(\frac{419}{429}\right)\) | \(e\left(\frac{49}{429}\right)\) | \(e\left(\frac{86}{143}\right)\) | \(e\left(\frac{138}{143}\right)\) | \(e\left(\frac{97}{429}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{353}{429}\right)\) | \(e\left(\frac{23}{39}\right)\) |
| \(\chi_{6013}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{429}\right)\) | \(e\left(\frac{368}{429}\right)\) | \(e\left(\frac{17}{429}\right)\) | \(e\left(\frac{217}{429}\right)\) | \(e\left(\frac{54}{143}\right)\) | \(e\left(\frac{80}{143}\right)\) | \(e\left(\frac{307}{429}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{215}{429}\right)\) | \(e\left(\frac{35}{39}\right)\) |
| \(\chi_{6013}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{429}\right)\) | \(e\left(\frac{326}{429}\right)\) | \(e\left(\frac{92}{429}\right)\) | \(e\left(\frac{64}{429}\right)\) | \(e\left(\frac{124}{143}\right)\) | \(e\left(\frac{46}{143}\right)\) | \(e\left(\frac{223}{429}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{356}{429}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{6013}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{290}{429}\right)\) | \(e\left(\frac{190}{429}\right)\) | \(e\left(\frac{151}{429}\right)\) | \(e\left(\frac{161}{429}\right)\) | \(e\left(\frac{17}{143}\right)\) | \(e\left(\frac{4}{143}\right)\) | \(e\left(\frac{380}{429}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{118}{429}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{6013}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{429}\right)\) | \(e\left(\frac{301}{429}\right)\) | \(e\left(\frac{106}{429}\right)\) | \(e\left(\frac{167}{429}\right)\) | \(e\left(\frac{118}{143}\right)\) | \(e\left(\frac{53}{143}\right)\) | \(e\left(\frac{173}{429}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{205}{429}\right)\) | \(e\left(\frac{37}{39}\right)\) |
| \(\chi_{6013}(471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{429}\right)\) | \(e\left(\frac{74}{429}\right)\) | \(e\left(\frac{113}{429}\right)\) | \(e\left(\frac{4}{429}\right)\) | \(e\left(\frac{115}{143}\right)\) | \(e\left(\frac{128}{143}\right)\) | \(e\left(\frac{148}{429}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{344}{429}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{6013}(480,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{386}{429}\right)\) | \(e\left(\frac{31}{429}\right)\) | \(e\left(\frac{343}{429}\right)\) | \(e\left(\frac{164}{429}\right)\) | \(e\left(\frac{139}{143}\right)\) | \(e\left(\frac{100}{143}\right)\) | \(e\left(\frac{62}{429}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{376}{429}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{6013}(494,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{429}\right)\) | \(e\left(\frac{235}{429}\right)\) | \(e\left(\frac{40}{429}\right)\) | \(e\left(\frac{233}{429}\right)\) | \(e\left(\frac{85}{143}\right)\) | \(e\left(\frac{20}{143}\right)\) | \(e\left(\frac{41}{429}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{304}{429}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{6013}(499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{429}\right)\) | \(e\left(\frac{200}{429}\right)\) | \(e\left(\frac{317}{429}\right)\) | \(e\left(\frac{34}{429}\right)\) | \(e\left(\frac{48}{143}\right)\) | \(e\left(\frac{87}{143}\right)\) | \(e\left(\frac{400}{429}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{350}{429}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{6013}(508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{429}\right)\) | \(e\left(\frac{370}{429}\right)\) | \(e\left(\frac{136}{429}\right)\) | \(e\left(\frac{20}{429}\right)\) | \(e\left(\frac{3}{143}\right)\) | \(e\left(\frac{68}{143}\right)\) | \(e\left(\frac{311}{429}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{4}{429}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{6013}(522,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{429}\right)\) | \(e\left(\frac{274}{429}\right)\) | \(e\left(\frac{1}{429}\right)\) | \(e\left(\frac{38}{429}\right)\) | \(e\left(\frac{20}{143}\right)\) | \(e\left(\frac{72}{143}\right)\) | \(e\left(\frac{119}{429}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{265}{429}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{6013}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{429}\right)\) | \(e\left(\frac{38}{429}\right)\) | \(e\left(\frac{116}{429}\right)\) | \(e\left(\frac{118}{429}\right)\) | \(e\left(\frac{32}{143}\right)\) | \(e\left(\frac{58}{143}\right)\) | \(e\left(\frac{76}{429}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{281}{429}\right)\) | \(e\left(\frac{14}{39}\right)\) |
| \(\chi_{6013}(571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{429}\right)\) | \(e\left(\frac{373}{429}\right)\) | \(e\left(\frac{100}{429}\right)\) | \(e\left(\frac{368}{429}\right)\) | \(e\left(\frac{141}{143}\right)\) | \(e\left(\frac{50}{143}\right)\) | \(e\left(\frac{317}{429}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{331}{429}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{6013}(585,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{404}{429}\right)\) | \(e\left(\frac{28}{429}\right)\) | \(e\left(\frac{379}{429}\right)\) | \(e\left(\frac{245}{429}\right)\) | \(e\left(\frac{1}{143}\right)\) | \(e\left(\frac{118}{143}\right)\) | \(e\left(\frac{56}{429}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{49}{429}\right)\) | \(e\left(\frac{37}{39}\right)\) |
| \(\chi_{6013}(604,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{429}\right)\) | \(e\left(\frac{215}{429}\right)\) | \(e\left(\frac{137}{429}\right)\) | \(e\left(\frac{58}{429}\right)\) | \(e\left(\frac{23}{143}\right)\) | \(e\left(\frac{140}{143}\right)\) | \(e\left(\frac{1}{429}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{269}{429}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{6013}(618,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{429}\right)\) | \(e\left(\frac{86}{429}\right)\) | \(e\left(\frac{398}{429}\right)\) | \(e\left(\frac{109}{429}\right)\) | \(e\left(\frac{95}{143}\right)\) | \(e\left(\frac{56}{143}\right)\) | \(e\left(\frac{172}{429}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{365}{429}\right)\) | \(e\left(\frac{5}{39}\right)\) |