from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51714, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([312,548,39]))
chi.galois_orbit()
[g,chi] = znchar(Mod(71,51714))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(51714\) | |
| Conductor: | \(8619\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(624\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 8619.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: | $\Q(\zeta_{624})$ |
| Fixed field: | Number field defined by a degree 624 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{51714}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{208}\right)\) | \(e\left(\frac{409}{624}\right)\) | \(e\left(\frac{245}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{275}{624}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{419}{624}\right)\) |
| \(\chi_{51714}(215,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{208}\right)\) | \(e\left(\frac{415}{624}\right)\) | \(e\left(\frac{35}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{51}{104}\right)\) | \(e\left(\frac{485}{624}\right)\) | \(e\left(\frac{119}{208}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{149}{624}\right)\) |
| \(\chi_{51714}(449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{208}\right)\) | \(e\left(\frac{115}{624}\right)\) | \(e\left(\frac{551}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{593}{624}\right)\) | \(e\left(\frac{43}{208}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{545}{624}\right)\) |
| \(\chi_{51714}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{208}\right)\) | \(e\left(\frac{229}{624}\right)\) | \(e\left(\frac{305}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{73}{104}\right)\) | \(e\left(\frac{215}{624}\right)\) | \(e\left(\frac{205}{208}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{407}{624}\right)\) |
| \(\chi_{51714}(1727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{208}\right)\) | \(e\left(\frac{395}{624}\right)\) | \(e\left(\frac{319}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{31}{104}\right)\) | \(e\left(\frac{409}{624}\right)\) | \(e\left(\frac{3}{208}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{217}{624}\right)\) |
| \(\chi_{51714}(1943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{208}\right)\) | \(e\left(\frac{157}{624}\right)\) | \(e\left(\frac{329}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{191}{624}\right)\) | \(e\left(\frac{37}{208}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{527}{624}\right)\) |
| \(\chi_{51714}(1961,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{119}{624}\right)\) | \(e\left(\frac{619}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{67}{104}\right)\) | \(e\left(\frac{109}{624}\right)\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{157}{624}\right)\) |
| \(\chi_{51714}(2069,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{208}\right)\) | \(e\left(\frac{149}{624}\right)\) | \(e\left(\frac{193}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{535}{624}\right)\) | \(e\left(\frac{157}{208}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{55}{624}\right)\) |
| \(\chi_{51714}(2411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{289}{624}\right)\) | \(e\left(\frac{77}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{443}{624}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{203}{624}\right)\) |
| \(\chi_{51714}(2663,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{383}{624}\right)\) | \(e\left(\frac{115}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{104}\right)\) | \(e\left(\frac{613}{624}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{133}{624}\right)\) |
| \(\chi_{51714}(2897,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{208}\right)\) | \(e\left(\frac{419}{624}\right)\) | \(e\left(\frac{103}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{55}{104}\right)\) | \(e\left(\frac{1}{624}\right)\) | \(e\left(\frac{59}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{385}{624}\right)\) |
| \(\chi_{51714}(3257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{191}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{103}{104}\right)\) | \(e\left(\frac{329}{624}\right)\) | \(e\left(\frac{67}{208}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{617}{624}\right)\) |
| \(\chi_{51714}(3473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{208}\right)\) | \(e\left(\frac{557}{624}\right)\) | \(e\left(\frac{265}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{463}{624}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{415}{624}\right)\) |
| \(\chi_{51714}(3491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{208}\right)\) | \(e\left(\frac{583}{624}\right)\) | \(e\left(\frac{395}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{104}\right)\) | \(e\left(\frac{125}{624}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{77}{624}\right)\) |
| \(\chi_{51714}(3941,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{208}\right)\) | \(e\left(\frac{17}{624}\right)\) | \(e\left(\frac{445}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{69}{104}\right)\) | \(e\left(\frac{283}{624}\right)\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{379}{624}\right)\) |
| \(\chi_{51714}(4049,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{361}{624}\right)\) | \(e\left(\frac{53}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{101}{104}\right)\) | \(e\left(\frac{467}{624}\right)\) | \(e\left(\frac{97}{208}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{83}{624}\right)\) |
| \(\chi_{51714}(4193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{208}\right)\) | \(e\left(\frac{463}{624}\right)\) | \(e\left(\frac{227}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{293}{624}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{485}{624}\right)\) |
| \(\chi_{51714}(4427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{163}{624}\right)\) | \(e\left(\frac{119}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{401}{624}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{257}{624}\right)\) |
| \(\chi_{51714}(4517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{208}\right)\) | \(e\left(\frac{181}{624}\right)\) | \(e\left(\frac{113}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{25}{104}\right)\) | \(e\left(\frac{407}{624}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{71}{624}\right)\) |
| \(\chi_{51714}(5579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{233}{624}\right)\) | \(e\left(\frac{373}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{77}{104}\right)\) | \(e\left(\frac{355}{624}\right)\) | \(e\left(\frac{145}{208}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{19}{624}\right)\) |
| \(\chi_{51714}(5705,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{539}{624}\right)\) | \(e\left(\frac{271}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{71}{104}\right)\) | \(e\left(\frac{457}{624}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{601}{624}\right)\) |
| \(\chi_{51714}(5921,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{109}{624}\right)\) | \(e\left(\frac{137}{624}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{57}{104}\right)\) | \(e\left(\frac{383}{624}\right)\) | \(e\left(\frac{133}{208}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{191}{624}\right)\) |
| \(\chi_{51714}(5939,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{263}{624}\right)\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{3}{104}\right)\) | \(e\left(\frac{157}{624}\right)\) | \(e\left(\frac{111}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{541}{624}\right)\) |
| \(\chi_{51714}(6047,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{5}{624}\right)\) | \(e\left(\frac{241}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{57}{104}\right)\) | \(e\left(\frac{487}{624}\right)\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{295}{624}\right)\) |
| \(\chi_{51714}(6389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{189}{208}\right)\) | \(e\left(\frac{241}{624}\right)\) | \(e\left(\frac{509}{624}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{85}{104}\right)\) | \(e\left(\frac{11}{624}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{491}{624}\right)\) |
| \(\chi_{51714}(6641,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{208}\right)\) | \(e\left(\frac{527}{624}\right)\) | \(e\left(\frac{67}{624}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{37}{624}\right)\) | \(e\left(\frac{103}{208}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{517}{624}\right)\) |
| \(\chi_{51714}(6875,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{208}\right)\) | \(e\left(\frac{563}{624}\right)\) | \(e\left(\frac{55}{624}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{95}{104}\right)\) | \(e\left(\frac{49}{624}\right)\) | \(e\left(\frac{187}{208}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{145}{624}\right)\) |
| \(\chi_{51714}(7235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{619}{624}\right)\) | \(e\left(\frac{383}{624}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{47}{104}\right)\) | \(e\left(\frac{137}{624}\right)\) | \(e\left(\frac{179}{208}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{329}{624}\right)\) |
| \(\chi_{51714}(7451,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{208}\right)\) | \(e\left(\frac{413}{624}\right)\) | \(e\left(\frac{313}{624}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{49}{104}\right)\) | \(e\left(\frac{415}{624}\right)\) | \(e\left(\frac{149}{208}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{31}{624}\right)\) |
| \(\chi_{51714}(7469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{208}\right)\) | \(e\left(\frac{7}{624}\right)\) | \(e\left(\frac{587}{624}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{557}{624}\right)\) | \(e\left(\frac{207}{208}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{413}{624}\right)\) |
| \(\chi_{51714}(7919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{497}{624}\right)\) | \(e\left(\frac{493}{624}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{235}{624}\right)\) | \(e\left(\frac{137}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{619}{624}\right)\) |