from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25857, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([52,2,273]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,25857))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(25857\) | |
| Conductor: | \(25857\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(312\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{312})$ |
| Fixed field: | Number field defined by a degree 312 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{25857}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{305}{312}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{215}{312}\right)\) | \(e\left(\frac{99}{104}\right)\) | \(e\left(\frac{125}{312}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{103}{312}\right)\) | \(e\left(\frac{277}{312}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{139}{312}\right)\) | \(e\left(\frac{79}{104}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{97}{312}\right)\) | \(e\left(\frac{67}{312}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{37}{312}\right)\) | \(e\left(\frac{33}{104}\right)\) | \(e\left(\frac{7}{312}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{269}{312}\right)\) | \(e\left(\frac{263}{312}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{257}{312}\right)\) | \(e\left(\frac{69}{104}\right)\) | \(e\left(\frac{251}{312}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{263}{312}\right)\) | \(e\left(\frac{53}{312}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{155}{312}\right)\) | \(e\left(\frac{23}{104}\right)\) | \(e\left(\frac{257}{312}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(1181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{227}{312}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{293}{312}\right)\) | \(e\left(\frac{73}{104}\right)\) | \(e\left(\frac{47}{312}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(1787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{163}{312}\right)\) | \(e\left(\frac{193}{312}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{223}{312}\right)\) | \(e\left(\frac{19}{104}\right)\) | \(e\left(\frac{253}{312}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(1991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{131}{312}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{51}{104}\right)\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(2021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{223}{312}\right)\) | \(e\left(\frac{109}{312}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{307}{312}\right)\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{193}{312}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(2048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{289}{312}\right)\) | \(e\left(\frac{235}{312}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{181}{312}\right)\) | \(e\left(\frac{49}{104}\right)\) | \(e\left(\frac{127}{312}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(2225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{311}{312}\right)\) | \(e\left(\frac{173}{312}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{35}{312}\right)\) | \(e\left(\frac{79}{104}\right)\) | \(e\left(\frac{209}{312}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(2984,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{205}{312}\right)\) | \(e\left(\frac{103}{312}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{29}{104}\right)\) | \(e\left(\frac{211}{312}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(3170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{107}{312}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{101}{312}\right)\) | \(e\left(\frac{17}{104}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(3776,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{283}{312}\right)\) | \(e\left(\frac{25}{312}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{79}{312}\right)\) | \(e\left(\frac{3}{104}\right)\) | \(e\left(\frac{133}{312}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(3980,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{179}{312}\right)\) | \(e\left(\frac{233}{312}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{287}{312}\right)\) | \(e\left(\frac{3}{104}\right)\) | \(e\left(\frac{29}{312}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(4010,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{31}{312}\right)\) | \(e\left(\frac{253}{312}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{163}{312}\right)\) | \(e\left(\frac{47}{104}\right)\) | \(e\left(\frac{73}{312}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(4106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{173}{312}\right)\) | \(e\left(\frac{23}{312}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{185}{312}\right)\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{35}{312}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(4214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{47}{312}\right)\) | \(e\left(\frac{293}{312}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{227}{312}\right)\) | \(e\left(\frac{31}{104}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(4973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{85}{312}\right)\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{145}{312}\right)\) | \(e\left(\frac{45}{104}\right)\) | \(e\left(\frac{19}{312}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(5969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{227}{312}\right)\) | \(e\left(\frac{41}{312}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{167}{312}\right)\) | \(e\left(\frac{59}{104}\right)\) | \(e\left(\frac{293}{312}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(5999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{151}{312}\right)\) | \(e\left(\frac{85}{312}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{19}{312}\right)\) | \(e\left(\frac{31}{104}\right)\) | \(e\left(\frac{265}{312}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(6026,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{49}{312}\right)\) | \(e\left(\frac{259}{312}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{157}{312}\right)\) | \(e\left(\frac{81}{104}\right)\) | \(e\left(\frac{55}{312}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(6095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{125}{312}\right)\) | \(e\left(\frac{215}{312}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{305}{312}\right)\) | \(e\left(\frac{5}{104}\right)\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(6203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{101}{312}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{107}{312}\right)\) | \(e\left(\frac{87}{104}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(6962,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{277}{312}\right)\) | \(e\left(\frac{127}{312}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{289}{312}\right)\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{139}{312}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(7148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{17}{312}\right)\) | \(e\left(\frac{179}{312}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{29}{312}\right)\) | \(e\left(\frac{9}{104}\right)\) | \(e\left(\frac{191}{312}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(7754,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{211}{312}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{103}{312}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{205}{312}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(7958,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{275}{312}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{47}{312}\right)\) | \(e\left(\frac{11}{104}\right)\) | \(e\left(\frac{245}{312}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{25857}(7988,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{229}{312}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{187}{312}\right)\) | \(e\left(\frac{15}{104}\right)\) | \(e\left(\frac{145}{312}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(8015,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{241}{312}\right)\) | \(e\left(\frac{115}{312}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{301}{312}\right)\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{175}{312}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{25857}(8084,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{53}{104}\right)\) | \(e\left(\frac{131}{312}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |