from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([483,506,616]))
chi.galois_orbit()
[g,chi] = znchar(Mod(63,40310))
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.fg | 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: | $\Q(\zeta_{644})$ |
| Fixed field: | Number field defined by a degree 644 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{40310}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{255}{644}\right)\) | \(e\left(\frac{3}{644}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{393}{644}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{121}{644}\right)\) |
| \(\chi_{40310}(183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{644}\right)\) | \(e\left(\frac{191}{644}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{549}{644}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{163}{322}\right)\) | \(e\left(\frac{421}{644}\right)\) | \(e\left(\frac{405}{644}\right)\) |
| \(\chi_{40310}(323,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{644}\right)\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{289}{644}\right)\) | \(e\left(\frac{47}{92}\right)\) | \(e\left(\frac{65}{161}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{353}{644}\right)\) | \(e\left(\frac{361}{644}\right)\) |
| \(\chi_{40310}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{644}\right)\) | \(e\left(\frac{177}{644}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{3}{644}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{407}{644}\right)\) | \(e\left(\frac{55}{644}\right)\) |
| \(\chi_{40310}(497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{545}{644}\right)\) | \(e\left(\frac{461}{644}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{499}{644}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{507}{644}\right)\) | \(e\left(\frac{347}{644}\right)\) |
| \(\chi_{40310}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{587}{644}\right)\) | \(e\left(\frac{363}{644}\right)\) | \(e\left(\frac{265}{322}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{541}{644}\right)\) | \(e\left(\frac{31}{92}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{409}{644}\right)\) | \(e\left(\frac{473}{644}\right)\) |
| \(\chi_{40310}(647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{261}{644}\right)\) | \(e\left(\frac{541}{644}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{31}{644}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{127}{644}\right)\) | \(e\left(\frac{139}{644}\right)\) |
| \(\chi_{40310}(747,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{197}{644}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{47}{644}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{151}{644}\right)\) | \(e\left(\frac{3}{644}\right)\) |
| \(\chi_{40310}(963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{431}{644}\right)\) | \(e\left(\frac{543}{644}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{293}{644}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{37}{644}\right)\) | \(e\left(\frac{5}{644}\right)\) |
| \(\chi_{40310}(1037,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{159}{644}\right)\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{319}{644}\right)\) | \(e\left(\frac{339}{644}\right)\) |
| \(\chi_{40310}(1053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{499}{644}\right)\) | \(e\left(\frac{415}{644}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{159}{322}\right)\) | \(e\left(\frac{269}{644}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{209}{644}\right)\) |
| \(\chi_{40310}(1367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{359}{644}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{571}{644}\right)\) |
| \(\chi_{40310}(1733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{644}\right)\) | \(e\left(\frac{155}{644}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{265}{322}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{109}{644}\right)\) | \(e\left(\frac{241}{644}\right)\) |
| \(\chi_{40310}(1907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{235}{644}\right)\) | \(e\left(\frac{171}{644}\right)\) |
| \(\chi_{40310}(1923,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{644}\right)\) | \(e\left(\frac{527}{644}\right)\) | \(e\left(\frac{37}{322}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{129}{644}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{24}{161}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{433}{644}\right)\) |
| \(\chi_{40310}(2023,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{644}\right)\) | \(e\left(\frac{351}{644}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{305}{644}\right)\) | \(e\left(\frac{633}{644}\right)\) |
| \(\chi_{40310}(2197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{617}{644}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{327}{644}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{117}{161}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{199}{644}\right)\) |
| \(\chi_{40310}(2353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{83}{644}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{569}{644}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{129}{644}\right)\) | \(e\left(\frac{557}{644}\right)\) |
| \(\chi_{40310}(2557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{521}{644}\right)\) | \(e\left(\frac{241}{644}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{275}{644}\right)\) |
| \(\chi_{40310}(2677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{644}\right)\) | \(e\left(\frac{625}{644}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{87}{644}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{211}{644}\right)\) | \(e\left(\frac{307}{644}\right)\) |
| \(\chi_{40310}(2963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{644}\right)\) | \(e\left(\frac{283}{644}\right)\) | \(e\left(\frac{227}{322}\right)\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{145}{644}\right)\) | \(e\left(\frac{37}{644}\right)\) |
| \(\chi_{40310}(3137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{644}\right)\) | \(e\left(\frac{269}{644}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{463}{644}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{131}{644}\right)\) | \(e\left(\frac{331}{644}\right)\) |
| \(\chi_{40310}(3183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{351}{644}\right)\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{397}{644}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{193}{644}\right)\) | \(e\left(\frac{409}{644}\right)\) |
| \(\chi_{40310}(3203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{607}{644}\right)\) | \(e\left(\frac{439}{644}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{65}{322}\right)\) | \(e\left(\frac{193}{644}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{209}{644}\right)\) | \(e\left(\frac{533}{644}\right)\) |
| \(\chi_{40310}(3427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{353}{644}\right)\) | \(e\left(\frac{633}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{137}{322}\right)\) | \(e\left(\frac{491}{644}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{495}{644}\right)\) | \(e\left(\frac{415}{644}\right)\) |
| \(\chi_{40310}(3833,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{591}{644}\right)\) | \(e\left(\frac{507}{644}\right)\) | \(e\left(\frac{269}{322}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{227}{322}\right)\) | \(e\left(\frac{369}{644}\right)\) | \(e\left(\frac{485}{644}\right)\) |
| \(\chi_{40310}(3937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{644}\right)\) | \(e\left(\frac{169}{644}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{243}{644}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{123}{644}\right)\) | \(e\left(\frac{591}{644}\right)\) |
| \(\chi_{40310}(3957,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{201}{644}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{219}{322}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{379}{644}\right)\) |
| \(\chi_{40310}(4037,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{644}\right)\) | \(e\left(\frac{25}{644}\right)\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{55}{644}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{163}{644}\right)\) | \(e\left(\frac{579}{644}\right)\) |
| \(\chi_{40310}(4227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{449}{644}\right)\) | \(e\left(\frac{225}{644}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{495}{644}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{179}{644}\right)\) | \(e\left(\frac{59}{644}\right)\) |
| \(\chi_{40310}(4247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{397}{644}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{487}{644}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{127}{644}\right)\) |