from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25857, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([0,40,117]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,25857))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(25857\) | |
| Conductor: | \(2873\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(624\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2873.ct | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{25857}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{312}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{575}{624}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{127}{624}\right)\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{127}{208}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{25857}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{312}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{161}{208}\right)\) | \(e\left(\frac{109}{624}\right)\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{125}{624}\right)\) | \(e\left(\frac{449}{624}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{25857}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{312}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{529}{624}\right)\) | \(e\left(\frac{73}{104}\right)\) | \(e\left(\frac{17}{624}\right)\) | \(e\left(\frac{101}{624}\right)\) | \(e\left(\frac{17}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{25857}(244,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{312}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{443}{624}\right)\) | \(e\left(\frac{11}{104}\right)\) | \(e\left(\frac{571}{624}\right)\) | \(e\left(\frac{199}{624}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{25857}(550,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{312}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{151}{208}\right)\) | \(e\left(\frac{619}{624}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{395}{624}\right)\) | \(e\left(\frac{71}{624}\right)\) | \(e\left(\frac{187}{208}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{25857}(946,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{312}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{47}{208}\right)\) | \(e\left(\frac{515}{624}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{499}{624}\right)\) | \(e\left(\frac{175}{624}\right)\) | \(e\left(\frac{83}{208}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{25857}(1180,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{312}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{19}{208}\right)\) | \(e\left(\frac{71}{624}\right)\) | \(e\left(\frac{79}{104}\right)\) | \(e\left(\frac{7}{624}\right)\) | \(e\left(\frac{115}{624}\right)\) | \(e\left(\frac{7}{208}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{25857}(1252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{312}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{175}{208}\right)\) | \(e\left(\frac{19}{624}\right)\) | \(e\left(\frac{27}{104}\right)\) | \(e\left(\frac{371}{624}\right)\) | \(e\left(\frac{479}{624}\right)\) | \(e\left(\frac{163}{208}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{25857}(1297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{312}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{473}{624}\right)\) | \(e\left(\frac{81}{104}\right)\) | \(e\left(\frac{73}{624}\right)\) | \(e\left(\frac{397}{624}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{25857}(1414,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{312}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{121}{208}\right)\) | \(e\left(\frac{485}{624}\right)\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{373}{624}\right)\) | \(e\left(\frac{601}{624}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{25857}(1486,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{131}{208}\right)\) | \(e\left(\frac{391}{624}\right)\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{311}{624}\right)\) | \(e\left(\frac{563}{624}\right)\) | \(e\left(\frac{103}{208}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{25857}(1603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{312}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{265}{624}\right)\) | \(e\left(\frac{81}{104}\right)\) | \(e\left(\frac{281}{624}\right)\) | \(e\left(\frac{605}{624}\right)\) | \(e\left(\frac{73}{208}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{25857}(1720,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{312}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{9}{208}\right)\) | \(e\left(\frac{373}{624}\right)\) | \(e\left(\frac{21}{104}\right)\) | \(e\left(\frac{485}{624}\right)\) | \(e\left(\frac{569}{624}\right)\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{25857}(1765,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{312}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{365}{624}\right)\) | \(e\left(\frac{37}{104}\right)\) | \(e\left(\frac{493}{624}\right)\) | \(e\left(\frac{433}{624}\right)\) | \(e\left(\frac{77}{208}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{25857}(1999,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{312}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{155}{208}\right)\) | \(e\left(\frac{623}{624}\right)\) | \(e\left(\frac{15}{104}\right)\) | \(e\left(\frac{79}{624}\right)\) | \(e\left(\frac{139}{624}\right)\) | \(e\left(\frac{79}{208}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{25857}(2071,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{312}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{177}{208}\right)\) | \(e\left(\frac{541}{624}\right)\) | \(e\left(\frac{101}{104}\right)\) | \(e\left(\frac{317}{624}\right)\) | \(e\left(\frac{305}{624}\right)\) | \(e\left(\frac{109}{208}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{25857}(2188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{312}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{181}{208}\right)\) | \(e\left(\frac{337}{624}\right)\) | \(e\left(\frac{41}{104}\right)\) | \(e\left(\frac{209}{624}\right)\) | \(e\left(\frac{581}{624}\right)\) | \(e\left(\frac{1}{208}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{25857}(2233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{312}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{23}{208}\right)\) | \(e\left(\frac{491}{624}\right)\) | \(e\left(\frac{19}{104}\right)\) | \(e\left(\frac{523}{624}\right)\) | \(e\left(\frac{391}{624}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{25857}(2305,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{312}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{107}{208}\right)\) | \(e\left(\frac{367}{624}\right)\) | \(e\left(\frac{7}{104}\right)\) | \(e\left(\frac{335}{624}\right)\) | \(e\left(\frac{155}{624}\right)\) | \(e\left(\frac{127}{208}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{25857}(2539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{312}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{167}{208}\right)\) | \(e\left(\frac{427}{624}\right)\) | \(e\left(\frac{43}{104}\right)\) | \(e\left(\frac{587}{624}\right)\) | \(e\left(\frac{551}{624}\right)\) | \(e\left(\frac{171}{208}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{25857}(2935,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{95}{208}\right)\) | \(e\left(\frac{563}{624}\right)\) | \(e\left(\frac{83}{104}\right)\) | \(e\left(\frac{451}{624}\right)\) | \(e\left(\frac{367}{624}\right)\) | \(e\left(\frac{35}{208}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{25857}(3169,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{312}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{67}{208}\right)\) | \(e\left(\frac{119}{624}\right)\) | \(e\left(\frac{87}{104}\right)\) | \(e\left(\frac{583}{624}\right)\) | \(e\left(\frac{307}{624}\right)\) | \(e\left(\frac{167}{208}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{25857}(3241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{312}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{191}{208}\right)\) | \(e\left(\frac{451}{624}\right)\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{563}{624}\right)\) | \(e\left(\frac{335}{624}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{25857}(3286,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{312}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{157}{208}\right)\) | \(e\left(\frac{521}{624}\right)\) | \(e\left(\frac{89}{104}\right)\) | \(e\left(\frac{25}{624}\right)\) | \(e\left(\frac{589}{624}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{25857}(3475,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{312}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{147}{208}\right)\) | \(e\left(\frac{199}{624}\right)\) | \(e\left(\frac{31}{104}\right)\) | \(e\left(\frac{503}{624}\right)\) | \(e\left(\frac{419}{624}\right)\) | \(e\left(\frac{87}{208}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{25857}(3592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{312}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{125}{208}\right)\) | \(e\left(\frac{73}{624}\right)\) | \(e\left(\frac{49}{104}\right)\) | \(e\left(\frac{473}{624}\right)\) | \(e\left(\frac{461}{624}\right)\) | \(e\left(\frac{57}{208}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{25857}(3709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{312}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{181}{624}\right)\) | \(e\left(\frac{93}{104}\right)\) | \(e\left(\frac{53}{624}\right)\) | \(e\left(\frac{425}{624}\right)\) | \(e\left(\frac{53}{208}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{25857}(3754,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{312}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{49}{208}\right)\) | \(e\left(\frac{413}{624}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{445}{624}\right)\) | \(e\left(\frac{1}{624}\right)\) | \(e\left(\frac{29}{208}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{25857}(3871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{312}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{165}{208}\right)\) | \(e\left(\frac{113}{624}\right)\) | \(e\left(\frac{73}{104}\right)\) | \(e\left(\frac{433}{624}\right)\) | \(e\left(\frac{517}{624}\right)\) | \(e\left(\frac{17}{208}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{25857}(3988,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{312}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{203}{208}\right)\) | \(e\left(\frac{47}{624}\right)\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{31}{624}\right)\) | \(e\left(\frac{331}{624}\right)\) | \(e\left(\frac{31}{208}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{25857}(4060,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{312}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{193}{208}\right)\) | \(e\left(\frac{349}{624}\right)\) | \(e\left(\frac{69}{104}\right)\) | \(e\left(\frac{509}{624}\right)\) | \(e\left(\frac{161}{624}\right)\) | \(e\left(\frac{93}{208}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{17}{24}\right)\) |