from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(86190, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([0,156,428,429]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,86190))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(86190\) | |
| Conductor: | \(14365\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(624\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 14365.jw | 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: | $\Q(\zeta_{624})$ |
| Fixed field: | Number field defined by a degree 624 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{86190}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{624}\right)\) | \(e\left(\frac{287}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{545}{624}\right)\) | \(e\left(\frac{123}{208}\right)\) | \(e\left(\frac{317}{624}\right)\) | \(e\left(\frac{539}{624}\right)\) | \(e\left(\frac{251}{312}\right)\) | \(e\left(\frac{11}{52}\right)\) |
| \(\chi_{86190}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{355}{624}\right)\) | \(e\left(\frac{419}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{413}{624}\right)\) | \(e\left(\frac{31}{208}\right)\) | \(e\left(\frac{41}{624}\right)\) | \(e\left(\frac{239}{624}\right)\) | \(e\left(\frac{311}{312}\right)\) | \(e\left(\frac{7}{52}\right)\) |
| \(\chi_{86190}(643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{624}\right)\) | \(e\left(\frac{197}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{624}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{335}{624}\right)\) | \(e\left(\frac{233}{624}\right)\) | \(e\left(\frac{281}{312}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{86190}(1423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{624}\right)\) | \(e\left(\frac{89}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{119}{624}\right)\) | \(e\left(\frac{157}{208}\right)\) | \(e\left(\frac{107}{624}\right)\) | \(e\left(\frac{365}{624}\right)\) | \(e\left(\frac{5}{312}\right)\) | \(e\left(\frac{17}{52}\right)\) |
| \(\chi_{86190}(2173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{624}\right)\) | \(e\left(\frac{181}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{235}{624}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{463}{624}\right)\) | \(e\left(\frac{553}{624}\right)\) | \(e\left(\frac{217}{312}\right)\) | \(e\left(\frac{41}{52}\right)\) |
| \(\chi_{86190}(3547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{624}\right)\) | \(e\left(\frac{139}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{277}{624}\right)\) | \(e\left(\frac{119}{208}\right)\) | \(e\left(\frac{97}{624}\right)\) | \(e\left(\frac{535}{624}\right)\) | \(e\left(\frac{127}{312}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{86190}(3937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{624}\right)\) | \(e\left(\frac{535}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{505}{624}\right)\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{517}{624}\right)\) | \(e\left(\frac{259}{624}\right)\) | \(e\left(\frac{307}{312}\right)\) | \(e\left(\frac{35}{52}\right)\) |
| \(\chi_{86190}(4153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{529}{624}\right)\) | \(e\left(\frac{257}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{575}{624}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{323}{624}\right)\) | \(e\left(\frac{437}{624}\right)\) | \(e\left(\frac{53}{312}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{86190}(4933,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{541}{624}\right)\) | \(e\left(\frac{461}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{371}{624}\right)\) | \(e\left(\frac{49}{208}\right)\) | \(e\left(\frac{407}{624}\right)\) | \(e\left(\frac{257}{624}\right)\) | \(e\left(\frac{89}{312}\right)\) | \(e\left(\frac{1}{52}\right)\) |
| \(\chi_{86190}(5077,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{624}\right)\) | \(e\left(\frac{443}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{389}{624}\right)\) | \(e\left(\frac{71}{208}\right)\) | \(e\left(\frac{161}{624}\right)\) | \(e\left(\frac{71}{624}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{11}{52}\right)\) |
| \(\chi_{86190}(5107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{624}\right)\) | \(e\left(\frac{319}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{97}{624}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{61}{624}\right)\) | \(e\left(\frac{523}{624}\right)\) | \(e\left(\frac{67}{312}\right)\) | \(e\left(\frac{51}{52}\right)\) |
| \(\chi_{86190}(5467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{624}\right)\) | \(e\left(\frac{263}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{569}{624}\right)\) | \(e\left(\frac{83}{208}\right)\) | \(e\left(\frac{197}{624}\right)\) | \(e\left(\frac{83}{624}\right)\) | \(e\left(\frac{155}{312}\right)\) | \(e\left(\frac{7}{52}\right)\) |
| \(\chi_{86190}(5683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{624}\right)\) | \(e\left(\frac{49}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{367}{624}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{115}{624}\right)\) | \(e\left(\frac{229}{624}\right)\) | \(e\left(\frac{157}{312}\right)\) | \(e\left(\frac{45}{52}\right)\) |
| \(\chi_{86190}(6463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{624}\right)\) | \(e\left(\frac{349}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{67}{624}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{55}{624}\right)\) | \(e\left(\frac{1}{624}\right)\) | \(e\left(\frac{265}{312}\right)\) | \(e\left(\frac{17}{52}\right)\) |
| \(\chi_{86190}(6637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{415}{624}\right)\) | \(e\left(\frac{191}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{17}{624}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{461}{624}\right)\) | \(e\left(\frac{587}{624}\right)\) | \(e\left(\frac{179}{312}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{86190}(7027,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{624}\right)\) | \(e\left(\frac{323}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{509}{624}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{185}{624}\right)\) | \(e\left(\frac{287}{624}\right)\) | \(e\left(\frac{239}{312}\right)\) | \(e\left(\frac{43}{52}\right)\) |
| \(\chi_{86190}(7273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{624}\right)\) | \(e\left(\frac{293}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{539}{624}\right)\) | \(e\left(\frac{185}{208}\right)\) | \(e\left(\frac{191}{624}\right)\) | \(e\left(\frac{185}{624}\right)\) | \(e\left(\frac{41}{312}\right)\) | \(e\left(\frac{25}{52}\right)\) |
| \(\chi_{86190}(8053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{624}\right)\) | \(e\left(\frac{185}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{23}{624}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{587}{624}\right)\) | \(e\left(\frac{317}{624}\right)\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{33}{52}\right)\) |
| \(\chi_{86190}(8803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{624}\right)\) | \(e\left(\frac{469}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{201}{208}\right)\) | \(e\left(\frac{31}{624}\right)\) | \(e\left(\frac{409}{624}\right)\) | \(e\left(\frac{121}{312}\right)\) | \(e\left(\frac{37}{52}\right)\) |
| \(\chi_{86190}(9583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{624}\right)\) | \(e\left(\frac{457}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{583}{624}\right)\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{283}{624}\right)\) | \(e\left(\frac{493}{624}\right)\) | \(e\left(\frac{229}{312}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{86190}(10177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{395}{624}\right)\) | \(e\left(\frac{475}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{565}{624}\right)\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{529}{624}\right)\) | \(e\left(\frac{55}{624}\right)\) | \(e\left(\frac{223}{312}\right)\) | \(e\left(\frac{51}{52}\right)\) |
| \(\chi_{86190}(10783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{624}\right)\) | \(e\left(\frac{353}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{479}{624}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{179}{624}\right)\) | \(e\left(\frac{389}{624}\right)\) | \(e\left(\frac{125}{312}\right)\) | \(e\left(\frac{9}{52}\right)\) |
| \(\chi_{86190}(11563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{624}\right)\) | \(e\left(\frac{557}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{275}{624}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{263}{624}\right)\) | \(e\left(\frac{209}{624}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{17}{52}\right)\) |
| \(\chi_{86190}(11707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{347}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{485}{624}\right)\) | \(e\left(\frac{119}{208}\right)\) | \(e\left(\frac{305}{624}\right)\) | \(e\left(\frac{119}{624}\right)\) | \(e\left(\frac{23}{312}\right)\) | \(e\left(\frac{47}{52}\right)\) |
| \(\chi_{86190}(11737,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{624}\right)\) | \(e\left(\frac{31}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{385}{624}\right)\) | \(e\left(\frac{43}{208}\right)\) | \(e\left(\frac{493}{624}\right)\) | \(e\left(\frac{43}{624}\right)\) | \(e\left(\frac{163}{312}\right)\) | \(e\left(\frac{3}{52}\right)\) |
| \(\chi_{86190}(12097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{487}{624}\right)\) | \(e\left(\frac{167}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{41}{624}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{341}{624}\right)\) | \(e\left(\frac{131}{624}\right)\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{43}{52}\right)\) |
| \(\chi_{86190}(12127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{624}\right)\) | \(e\left(\frac{115}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{301}{624}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{601}{624}\right)\) | \(e\left(\frac{79}{624}\right)\) | \(e\left(\frac{31}{312}\right)\) | \(e\left(\frac{43}{52}\right)\) |
| \(\chi_{86190}(12313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{497}{624}\right)\) | \(e\left(\frac{337}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{79}{624}\right)\) | \(e\left(\frac{85}{208}\right)\) | \(e\left(\frac{307}{624}\right)\) | \(e\left(\frac{85}{624}\right)\) | \(e\left(\frac{61}{312}\right)\) | \(e\left(\frac{41}{52}\right)\) |
| \(\chi_{86190}(13267,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{624}\right)\) | \(e\left(\frac{95}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{113}{624}\right)\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{605}{624}\right)\) | \(e\left(\frac{11}{624}\right)\) | \(e\left(\frac{107}{312}\right)\) | \(e\left(\frac{31}{52}\right)\) |
| \(\chi_{86190}(13657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{624}\right)\) | \(e\left(\frac{227}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{605}{624}\right)\) | \(e\left(\frac{127}{208}\right)\) | \(e\left(\frac{329}{624}\right)\) | \(e\left(\frac{335}{624}\right)\) | \(e\left(\frac{167}{312}\right)\) | \(e\left(\frac{27}{52}\right)\) |
| \(\chi_{86190}(13903,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{624}\right)\) | \(e\left(\frac{389}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{443}{624}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{47}{624}\right)\) | \(e\left(\frac{137}{624}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{41}{52}\right)\) |