from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([161,207,588]))
chi.galois_orbit()
[g,chi] = znchar(Mod(77,40310))
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.fj | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{40310}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{275}{644}\right)\) | \(e\left(\frac{625}{644}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{355}{644}\right)\) | \(e\left(\frac{535}{644}\right)\) | \(e\left(\frac{121}{322}\right)\) |
| \(\chi_{40310}(403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{33}{644}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{419}{644}\right)\) | \(e\left(\frac{429}{644}\right)\) | \(e\left(\frac{193}{644}\right)\) | \(e\left(\frac{285}{322}\right)\) |
| \(\chi_{40310}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{579}{644}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{53}{644}\right)\) | \(e\left(\frac{179}{644}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{107}{644}\right)\) | \(e\left(\frac{45}{644}\right)\) | \(e\left(\frac{349}{644}\right)\) | \(e\left(\frac{165}{322}\right)\) |
| \(\chi_{40310}(1007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{429}{644}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{423}{644}\right)\) | \(e\left(\frac{493}{644}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{453}{644}\right)\) | \(e\left(\frac{347}{644}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{199}{322}\right)\) |
| \(\chi_{40310}(1017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{605}{644}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{547}{644}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{193}{644}\right)\) | \(e\left(\frac{27}{644}\right)\) | \(e\left(\frac{467}{644}\right)\) | \(e\left(\frac{99}{322}\right)\) |
| \(\chi_{40310}(1237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{377}{644}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{121}{644}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{383}{644}\right)\) | \(e\left(\frac{423}{644}\right)\) | \(e\left(\frac{9}{322}\right)\) |
| \(\chi_{40310}(1287,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{375}{644}\right)\) | \(e\left(\frac{501}{644}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{429}{644}\right)\) | \(e\left(\frac{367}{644}\right)\) | \(e\left(\frac{27}{644}\right)\) | \(e\left(\frac{165}{322}\right)\) |
| \(\chi_{40310}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{59}{644}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{477}{644}\right)\) | \(e\left(\frac{323}{644}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{319}{644}\right)\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{565}{644}\right)\) | \(e\left(\frac{197}{322}\right)\) |
| \(\chi_{40310}(1563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{15}{644}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{285}{644}\right)\) | \(e\left(\frac{355}{644}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{223}{644}\right)\) | \(e\left(\frac{485}{644}\right)\) | \(e\left(\frac{613}{644}\right)\) | \(e\left(\frac{61}{322}\right)\) |
| \(\chi_{40310}(1593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{227}{644}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{449}{644}\right)\) | \(e\left(\frac{435}{644}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{627}{644}\right)\) | \(e\left(\frac{41}{644}\right)\) | \(e\left(\frac{89}{644}\right)\) | \(e\left(\frac{43}{322}\right)\) |
| \(\chi_{40310}(1813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{347}{644}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{153}{644}\right)\) | \(e\left(\frac{55}{644}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{479}{644}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{485}{644}\right)\) | \(e\left(\frac{209}{322}\right)\) |
| \(\chi_{40310}(1887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{185}{644}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{295}{644}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{389}{644}\right)\) | \(e\left(\frac{615}{644}\right)\) | \(e\left(\frac{47}{644}\right)\) | \(e\left(\frac{1}{322}\right)\) |
| \(\chi_{40310}(2397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{153}{644}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{401}{644}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{439}{644}\right)\) | \(e\left(\frac{199}{644}\right)\) | \(e\left(\frac{107}{322}\right)\) |
| \(\chi_{40310}(2463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{141}{644}\right)\) | \(e\left(\frac{379}{644}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{151}{644}\right)\) | \(e\left(\frac{545}{644}\right)\) | \(e\left(\frac{5}{644}\right)\) | \(e\left(\frac{281}{322}\right)\) |
| \(\chi_{40310}(2747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{597}{644}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{395}{644}\right)\) | \(e\left(\frac{605}{644}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{117}{644}\right)\) | \(e\left(\frac{627}{644}\right)\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{45}{322}\right)\) |
| \(\chi_{40310}(2753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{479}{644}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{603}{644}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{123}{644}\right)\) | \(e\left(\frac{461}{644}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{295}{322}\right)\) |
| \(\chi_{40310}(2757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{297}{644}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{491}{644}\right)\) | \(e\left(\frac{589}{644}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{165}{644}\right)\) | \(e\left(\frac{587}{644}\right)\) | \(e\left(\frac{159}{644}\right)\) | \(e\left(\frac{113}{322}\right)\) |
| \(\chi_{40310}(3113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{195}{644}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{485}{644}\right)\) | \(e\left(\frac{107}{644}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{323}{644}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{241}{644}\right)\) | \(e\left(\frac{149}{322}\right)\) |
| \(\chi_{40310}(3303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{257}{644}\right)\) | \(e\left(\frac{467}{644}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{531}{644}\right)\) | \(e\left(\frac{121}{644}\right)\) | \(e\left(\frac{137}{644}\right)\) | \(e\left(\frac{229}{322}\right)\) |
| \(\chi_{40310}(3693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{223}{644}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{373}{644}\right)\) | \(e\left(\frac{555}{644}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{267}{644}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{269}{644}\right)\) | \(e\left(\frac{177}{322}\right)\) |
| \(\chi_{40310}(3947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{439}{644}\right)\) | \(e\left(\frac{61}{644}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{461}{644}\right)\) | \(e\left(\frac{555}{644}\right)\) | \(e\left(\frac{11}{644}\right)\) | \(e\left(\frac{103}{322}\right)\) |
| \(\chi_{40310}(3983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{587}{644}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{205}{644}\right)\) | \(e\left(\frac{583}{644}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{183}{644}\right)\) | \(e\left(\frac{89}{644}\right)\) | \(e\left(\frac{633}{644}\right)\) | \(e\left(\frac{219}{322}\right)\) |
| \(\chi_{40310}(4137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{303}{644}\right)\) | \(e\left(\frac{513}{644}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{393}{644}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{367}{644}\right)\) | \(e\left(\frac{275}{322}\right)\) |
| \(\chi_{40310}(4493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{31}{644}\right)\) | \(e\left(\frac{110}{161}\right)\) | \(e\left(\frac{589}{644}\right)\) | \(e\left(\frac{519}{644}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{375}{644}\right)\) | \(e\left(\frac{573}{644}\right)\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{169}{322}\right)\) |
| \(\chi_{40310}(4503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{471}{644}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{577}{644}\right)\) | \(e\left(\frac{199}{644}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{47}{644}\right)\) | \(e\left(\frac{417}{644}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{241}{322}\right)\) |
| \(\chi_{40310}(4513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{523}{644}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{277}{644}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{219}{644}\right)\) | \(e\left(\frac{381}{644}\right)\) | \(e\left(\frac{293}{644}\right)\) | \(e\left(\frac{109}{322}\right)\) |
| \(\chi_{40310}(4527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{327}{644}\right)\) | \(e\left(\frac{509}{644}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{387}{644}\right)\) | \(e\left(\frac{39}{644}\right)\) | \(e\left(\frac{131}{322}\right)\) |
| \(\chi_{40310}(4783,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{87}{644}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{365}{644}\right)\) | \(e\left(\frac{127}{644}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{263}{644}\right)\) | \(e\left(\frac{237}{644}\right)\) | \(e\left(\frac{593}{644}\right)\) | \(e\left(\frac{225}{322}\right)\) |
| \(\chi_{40310}(4803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{322}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{137}{644}\right)\) | \(e\left(\frac{487}{644}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{471}{644}\right)\) | \(e\left(\frac{493}{644}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{305}{322}\right)\) |
| \(\chi_{40310}(4817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{449}{644}\right)\) | \(e\left(\frac{4}{161}\right)\) | \(e\left(\frac{159}{644}\right)\) | \(e\left(\frac{537}{644}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{135}{644}\right)\) | \(e\left(\frac{403}{644}\right)\) | \(e\left(\frac{173}{322}\right)\) |
| \(\chi_{40310}(4857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{585}{644}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{167}{644}\right)\) | \(e\left(\frac{321}{644}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{325}{644}\right)\) | \(e\left(\frac{239}{644}\right)\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{125}{322}\right)\) |