from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([322,529,490]))
chi.galois_orbit()
[g,chi] = znchar(Mod(39,40310))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(40310\) | |
| Conductor: | \(20155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(644\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 20155.fb | 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: | $\Q(\zeta_{644})$ |
| Fixed field: | Number field defined by a degree 644 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{40310}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{517}{644}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{233}{644}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{519}{644}\right)\) | \(e\left(\frac{131}{644}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{263}{644}\right)\) |
| \(\chi_{40310}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{644}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{563}{644}\right)\) | \(e\left(\frac{631}{644}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{495}{644}\right)\) |
| \(\chi_{40310}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{283}{644}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{61}{644}\right)\) | \(e\left(\frac{137}{644}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{629}{644}\right)\) |
| \(\chi_{40310}(769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{644}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{461}{644}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{311}{644}\right)\) | \(e\left(\frac{519}{644}\right)\) | \(e\left(\frac{102}{161}\right)\) | \(e\left(\frac{103}{644}\right)\) |
| \(\chi_{40310}(909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{541}{644}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{351}{644}\right)\) | \(e\left(\frac{271}{644}\right)\) | \(e\left(\frac{97}{161}\right)\) | \(e\left(\frac{431}{644}\right)\) |
| \(\chi_{40310}(939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{355}{644}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{515}{644}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{177}{644}\right)\) | \(e\left(\frac{577}{644}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{421}{644}\right)\) |
| \(\chi_{40310}(1139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{269}{644}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{215}{644}\right)\) | \(e\left(\frac{599}{644}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{475}{644}\right)\) |
| \(\chi_{40310}(1199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{461}{644}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{569}{644}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{43}{644}\right)\) | \(e\left(\frac{635}{644}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{95}{644}\right)\) |
| \(\chi_{40310}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{325}{644}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{243}{644}\right)\) | \(e\left(\frac{39}{644}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{447}{644}\right)\) |
| \(\chi_{40310}(1539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{339}{644}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{519}{644}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{501}{644}\right)\) | \(e\left(\frac{629}{644}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{373}{644}\right)\) |
| \(\chi_{40310}(1969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{589}{644}\right)\) | \(e\left(\frac{137}{322}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{27}{644}\right)\) | \(e\left(\frac{219}{644}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{479}{644}\right)\) |
| \(\chi_{40310}(2049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{363}{644}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{191}{644}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{337}{644}\right)\) | \(e\left(\frac{229}{644}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{445}{644}\right)\) |
| \(\chi_{40310}(2099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{263}{644}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{373}{644}\right)\) | \(e\left(\frac{521}{644}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{225}{644}\right)\) |
| \(\chi_{40310}(2299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{644}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{633}{644}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{179}{644}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{615}{644}\right)\) |
| \(\chi_{40310}(2439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{644}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{150}{161}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{293}{644}\right)\) | \(e\left(\frac{373}{644}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{213}{644}\right)\) |
| \(\chi_{40310}(2589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{644}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{17}{644}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{411}{644}\right)\) | \(e\left(\frac{543}{644}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{279}{644}\right)\) |
| \(\chi_{40310}(2649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{121}{644}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{463}{644}\right)\) | \(e\left(\frac{607}{644}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{319}{644}\right)\) |
| \(\chi_{40310}(2689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{583}{644}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{229}{644}\right)\) | \(e\left(\frac{641}{644}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{461}{644}\right)\) |
| \(\chi_{40310}(2839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{251}{644}\right)\) | \(e\left(\frac{247}{644}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{255}{644}\right)\) |
| \(\chi_{40310}(2979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{555}{644}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{313}{644}\right)\) | \(e\left(\frac{249}{644}\right)\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{377}{644}\right)\) |
| \(\chi_{40310}(3259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{579}{644}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{459}{644}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{493}{644}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{449}{644}\right)\) |
| \(\chi_{40310}(3279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{644}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{43}{644}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{585}{644}\right)\) | \(e\left(\frac{237}{644}\right)\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{289}{644}\right)\) |
| \(\chi_{40310}(3469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{285}{644}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{613}{644}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{387}{644}\right)\) | \(e\left(\frac{563}{644}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{211}{644}\right)\) |
| \(\chi_{40310}(3549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{644}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{599}{644}\right)\) | \(e\left(\frac{22}{161}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{541}{644}\right)\) | \(e\left(\frac{381}{644}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{57}{644}\right)\) |
| \(\chi_{40310}(3559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{644}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{255}{644}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{369}{644}\right)\) | \(e\left(\frac{417}{644}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{321}{644}\right)\) |
| \(\chi_{40310}(3569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{644}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{267}{644}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{53}{644}\right)\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{177}{644}\right)\) |
| \(\chi_{40310}(3709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{593}{644}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{55}{644}\right)\) | \(e\left(\frac{303}{644}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{451}{644}\right)\) |
| \(\chi_{40310}(4039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{597}{644}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{213}{644}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{515}{644}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{503}{644}\right)\) |
| \(\chi_{40310}(4079,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{527}{644}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{505}{644}\right)\) | \(e\left(\frac{89}{644}\right)\) | \(e\left(\frac{118}{161}\right)\) | \(e\left(\frac{277}{644}\right)\) |
| \(\chi_{40310}(4319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{644}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{197}{644}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{179}{644}\right)\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{51}{644}\right)\) |
| \(\chi_{40310}(4369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{279}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{51}{644}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{589}{644}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{193}{644}\right)\) |