from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([322,391,280]))
chi.galois_orbit()
[g,chi] = znchar(Mod(79,40310))
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.fc | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{40310}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{644}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{635}{644}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{55}{644}\right)\) |
| \(\chi_{40310}(369,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{261}{644}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{551}{644}\right)\) | \(e\left(\frac{319}{644}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{139}{644}\right)\) |
| \(\chi_{40310}(409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{505}{644}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{279}{644}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{227}{644}\right)\) |
| \(\chi_{40310}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{531}{644}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{209}{322}\right)\) | \(e\left(\frac{149}{644}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{477}{644}\right)\) | \(e\left(\frac{5}{644}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{305}{644}\right)\) |
| \(\chi_{40310}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{641}{644}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{363}{644}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{423}{644}\right)\) | \(e\left(\frac{211}{644}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{635}{644}\right)\) |
| \(\chi_{40310}(889,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{644}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{163}{644}\right)\) | \(e\left(\frac{67}{161}\right)\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{323}{644}\right)\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{459}{644}\right)\) |
| \(\chi_{40310}(959,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{644}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{631}{644}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{235}{644}\right)\) | \(e\left(\frac{475}{644}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{639}{644}\right)\) |
| \(\chi_{40310}(1469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{601}{644}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{51}{644}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{267}{644}\right)\) | \(e\left(\frac{19}{644}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{515}{644}\right)\) |
| \(\chi_{40310}(1519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{644}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{151}{644}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{31}{644}\right)\) | \(e\left(\frac{283}{322}\right)\) | \(e\left(\frac{603}{644}\right)\) |
| \(\chi_{40310}(1609,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{644}\right)\) | \(e\left(\frac{219}{322}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{111}{644}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{155}{644}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{439}{644}\right)\) |
| \(\chi_{40310}(1759,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{629}{644}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{527}{644}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{411}{644}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{599}{644}\right)\) |
| \(\chi_{40310}(1859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{523}{644}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{503}{644}\right)\) | \(e\left(\frac{359}{644}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{3}{644}\right)\) |
| \(\chi_{40310}(2009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{644}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{163}{322}\right)\) | \(e\left(\frac{241}{644}\right)\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{201}{644}\right)\) | \(e\left(\frac{557}{644}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{489}{644}\right)\) |
| \(\chi_{40310}(2119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{493}{644}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{39}{644}\right)\) | \(e\left(\frac{531}{644}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{191}{644}\right)\) |
| \(\chi_{40310}(2149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{495}{644}\right)\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{339}{644}\right)\) |
| \(\chi_{40310}(2289,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{644}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{81}{644}\right)\) | \(e\left(\frac{13}{644}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{149}{644}\right)\) |
| \(\chi_{40310}(2399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{644}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{283}{644}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{35}{92}\right)\) | \(e\left(\frac{383}{644}\right)\) | \(e\left(\frac{459}{644}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{307}{644}\right)\) |
| \(\chi_{40310}(2479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{644}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{61}{161}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{395}{644}\right)\) | \(e\left(\frac{127}{644}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{19}{644}\right)\) |
| \(\chi_{40310}(2579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{395}{644}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{505}{644}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{333}{644}\right)\) | \(e\left(\frac{125}{644}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{541}{644}\right)\) |
| \(\chi_{40310}(3019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{187}{644}\right)\) | \(e\left(\frac{66}{161}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{335}{644}\right)\) | \(e\left(\frac{499}{644}\right)\) | \(e\left(\frac{37}{322}\right)\) | \(e\left(\frac{171}{644}\right)\) |
| \(\chi_{40310}(3309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{131}{644}\right)\) | \(e\left(\frac{122}{161}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{307}{644}\right)\) | \(e\left(\frac{415}{644}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{199}{644}\right)\) |
| \(\chi_{40310}(3519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{644}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{317}{644}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{561}{644}\right)\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{221}{644}\right)\) |
| \(\chi_{40310}(3669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{291}{644}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{209}{644}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{185}{644}\right)\) | \(e\left(\frac{141}{644}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{229}{644}\right)\) |
| \(\chi_{40310}(3739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{535}{644}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{309}{644}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{557}{644}\right)\) | \(e\left(\frac{153}{644}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{317}{644}\right)\) |
| \(\chi_{40310}(3759,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{607}{644}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{613}{644}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{65}{644}\right)\) | \(e\left(\frac{241}{644}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{533}{644}\right)\) |
| \(\chi_{40310}(3789,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{644}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{191}{644}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{35}{92}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{551}{644}\right)\) | \(e\left(\frac{117}{322}\right)\) | \(e\left(\frac{123}{644}\right)\) |
| \(\chi_{40310}(3859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{605}{644}\right)\) | \(e\left(\frac{103}{322}\right)\) | \(e\left(\frac{283}{322}\right)\) | \(e\left(\frac{211}{644}\right)\) | \(e\left(\frac{65}{161}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{347}{644}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{527}{644}\right)\) |
| \(\chi_{40310}(4249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{644}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{97}{644}\right)\) | \(e\left(\frac{110}{161}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{129}{644}\right)\) | \(e\left(\frac{617}{644}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{285}{644}\right)\) |
| \(\chi_{40310}(4389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{644}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{65}{644}\right)\) | \(e\left(\frac{4}{161}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{201}{644}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{25}{644}\right)\) |
| \(\chi_{40310}(4539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{644}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{117}{161}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{45}{644}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{369}{644}\right)\) |
| \(\chi_{40310}(4579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{397}{644}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{601}{644}\right)\) | \(e\left(\frac{9}{644}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{549}{644}\right)\) |