from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1117, base_ring=CyclotomicField(1116))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1117))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1117\) | |
| Conductor: | \(1117\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1116\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1116})$ |
| Fixed field: | Number field defined by a degree 1116 polynomial (not computed) |
First 31 of 360 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1117}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1116}\right)\) | \(e\left(\frac{92}{93}\right)\) | \(e\left(\frac{1}{558}\right)\) | \(e\left(\frac{251}{372}\right)\) | \(e\left(\frac{1105}{1116}\right)\) | \(e\left(\frac{173}{558}\right)\) | \(e\left(\frac{1}{372}\right)\) | \(e\left(\frac{91}{93}\right)\) | \(e\left(\frac{377}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1105}{1116}\right)\) | \(e\left(\frac{11}{93}\right)\) | \(e\left(\frac{547}{558}\right)\) | \(e\left(\frac{215}{372}\right)\) | \(e\left(\frac{121}{1116}\right)\) | \(e\left(\frac{329}{558}\right)\) | \(e\left(\frac{361}{372}\right)\) | \(e\left(\frac{22}{93}\right)\) | \(e\left(\frac{317}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{1116}\right)\) | \(e\left(\frac{25}{93}\right)\) | \(e\left(\frac{347}{558}\right)\) | \(e\left(\frac{49}{372}\right)\) | \(e\left(\frac{647}{1116}\right)\) | \(e\left(\frac{325}{558}\right)\) | \(e\left(\frac{347}{372}\right)\) | \(e\left(\frac{50}{93}\right)\) | \(e\left(\frac{247}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{1116}\right)\) | \(e\left(\frac{73}{93}\right)\) | \(e\left(\frac{299}{558}\right)\) | \(e\left(\frac{277}{372}\right)\) | \(e\left(\frac{59}{1116}\right)\) | \(e\left(\frac{391}{558}\right)\) | \(e\left(\frac{299}{372}\right)\) | \(e\left(\frac{53}{93}\right)\) | \(e\left(\frac{7}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1093}{1116}\right)\) | \(e\left(\frac{23}{93}\right)\) | \(e\left(\frac{535}{558}\right)\) | \(e\left(\frac{179}{372}\right)\) | \(e\left(\frac{253}{1116}\right)\) | \(e\left(\frac{485}{558}\right)\) | \(e\left(\frac{349}{372}\right)\) | \(e\left(\frac{46}{93}\right)\) | \(e\left(\frac{257}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{755}{1116}\right)\) | \(e\left(\frac{82}{93}\right)\) | \(e\left(\frac{197}{558}\right)\) | \(e\left(\frac{157}{372}\right)\) | \(e\left(\frac{623}{1116}\right)\) | \(e\left(\frac{43}{558}\right)\) | \(e\left(\frac{11}{372}\right)\) | \(e\left(\frac{71}{93}\right)\) | \(e\left(\frac{55}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{613}{1116}\right)\) | \(e\left(\frac{38}{93}\right)\) | \(e\left(\frac{55}{558}\right)\) | \(e\left(\frac{227}{372}\right)\) | \(e\left(\frac{1069}{1116}\right)\) | \(e\left(\frac{29}{558}\right)\) | \(e\left(\frac{241}{372}\right)\) | \(e\left(\frac{76}{93}\right)\) | \(e\left(\frac{89}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{863}{1116}\right)\) | \(e\left(\frac{67}{93}\right)\) | \(e\left(\frac{305}{558}\right)\) | \(e\left(\frac{109}{372}\right)\) | \(e\left(\frac{551}{1116}\right)\) | \(e\left(\frac{313}{558}\right)\) | \(e\left(\frac{119}{372}\right)\) | \(e\left(\frac{41}{93}\right)\) | \(e\left(\frac{37}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{1116}\right)\) | \(e\left(\frac{88}{93}\right)\) | \(e\left(\frac{5}{558}\right)\) | \(e\left(\frac{139}{372}\right)\) | \(e\left(\frac{1061}{1116}\right)\) | \(e\left(\frac{307}{558}\right)\) | \(e\left(\frac{5}{372}\right)\) | \(e\left(\frac{83}{93}\right)\) | \(e\left(\frac{211}{558}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{1117}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1099}{1116}\right)\) | \(e\left(\frac{17}{93}\right)\) | \(e\left(\frac{541}{558}\right)\) | \(e\left(\frac{197}{372}\right)\) | \(e\left(\frac{187}{1116}\right)\) | \(e\left(\frac{407}{558}\right)\) | \(e\left(\frac{355}{372}\right)\) | \(e\left(\frac{34}{93}\right)\) | \(e\left(\frac{287}{558}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{1117}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{1116}\right)\) | \(e\left(\frac{37}{93}\right)\) | \(e\left(\frac{335}{558}\right)\) | \(e\left(\frac{13}{372}\right)\) | \(e\left(\frac{779}{1116}\right)\) | \(e\left(\frac{481}{558}\right)\) | \(e\left(\frac{335}{372}\right)\) | \(e\left(\frac{74}{93}\right)\) | \(e\left(\frac{187}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{1116}\right)\) | \(e\left(\frac{55}{93}\right)\) | \(e\left(\frac{503}{558}\right)\) | \(e\left(\frac{145}{372}\right)\) | \(e\left(\frac{47}{1116}\right)\) | \(e\left(\frac{529}{558}\right)\) | \(e\left(\frac{131}{372}\right)\) | \(e\left(\frac{17}{93}\right)\) | \(e\left(\frac{469}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{467}{1116}\right)\) | \(e\left(\frac{91}{93}\right)\) | \(e\left(\frac{467}{558}\right)\) | \(e\left(\frac{37}{372}\right)\) | \(e\left(\frac{443}{1116}\right)\) | \(e\left(\frac{439}{558}\right)\) | \(e\left(\frac{95}{372}\right)\) | \(e\left(\frac{89}{93}\right)\) | \(e\left(\frac{289}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{1116}\right)\) | \(e\left(\frac{74}{93}\right)\) | \(e\left(\frac{391}{558}\right)\) | \(e\left(\frac{305}{372}\right)\) | \(e\left(\frac{163}{1116}\right)\) | \(e\left(\frac{125}{558}\right)\) | \(e\left(\frac{19}{372}\right)\) | \(e\left(\frac{55}{93}\right)\) | \(e\left(\frac{95}{558}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{1117}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{1116}\right)\) | \(e\left(\frac{85}{93}\right)\) | \(e\left(\frac{287}{558}\right)\) | \(e\left(\frac{241}{372}\right)\) | \(e\left(\frac{191}{1116}\right)\) | \(e\left(\frac{547}{558}\right)\) | \(e\left(\frac{287}{372}\right)\) | \(e\left(\frac{77}{93}\right)\) | \(e\left(\frac{505}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1081}{1116}\right)\) | \(e\left(\frac{35}{93}\right)\) | \(e\left(\frac{523}{558}\right)\) | \(e\left(\frac{143}{372}\right)\) | \(e\left(\frac{385}{1116}\right)\) | \(e\left(\frac{83}{558}\right)\) | \(e\left(\frac{337}{372}\right)\) | \(e\left(\frac{70}{93}\right)\) | \(e\left(\frac{197}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{1116}\right)\) | \(e\left(\frac{23}{93}\right)\) | \(e\left(\frac{349}{558}\right)\) | \(e\left(\frac{179}{372}\right)\) | \(e\left(\frac{625}{1116}\right)\) | \(e\left(\frac{113}{558}\right)\) | \(e\left(\frac{349}{372}\right)\) | \(e\left(\frac{46}{93}\right)\) | \(e\left(\frac{443}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{683}{1116}\right)\) | \(e\left(\frac{61}{93}\right)\) | \(e\left(\frac{125}{558}\right)\) | \(e\left(\frac{313}{372}\right)\) | \(e\left(\frac{299}{1116}\right)\) | \(e\left(\frac{421}{558}\right)\) | \(e\left(\frac{311}{372}\right)\) | \(e\left(\frac{29}{93}\right)\) | \(e\left(\frac{253}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{743}{1116}\right)\) | \(e\left(\frac{1}{93}\right)\) | \(e\left(\frac{185}{558}\right)\) | \(e\left(\frac{121}{372}\right)\) | \(e\left(\frac{755}{1116}\right)\) | \(e\left(\frac{199}{558}\right)\) | \(e\left(\frac{371}{372}\right)\) | \(e\left(\frac{2}{93}\right)\) | \(e\left(\frac{553}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{581}{1116}\right)\) | \(e\left(\frac{70}{93}\right)\) | \(e\left(\frac{23}{558}\right)\) | \(e\left(\frac{7}{372}\right)\) | \(e\left(\frac{305}{1116}\right)\) | \(e\left(\frac{73}{558}\right)\) | \(e\left(\frac{209}{372}\right)\) | \(e\left(\frac{47}{93}\right)\) | \(e\left(\frac{301}{558}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{1117}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{1116}\right)\) | \(e\left(\frac{19}{93}\right)\) | \(e\left(\frac{167}{558}\right)\) | \(e\left(\frac{253}{372}\right)\) | \(e\left(\frac{395}{1116}\right)\) | \(e\left(\frac{433}{558}\right)\) | \(e\left(\frac{167}{372}\right)\) | \(e\left(\frac{38}{93}\right)\) | \(e\left(\frac{463}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{1116}\right)\) | \(e\left(\frac{71}{93}\right)\) | \(e\left(\frac{301}{558}\right)\) | \(e\left(\frac{35}{372}\right)\) | \(e\left(\frac{37}{1116}\right)\) | \(e\left(\frac{179}{558}\right)\) | \(e\left(\frac{301}{372}\right)\) | \(e\left(\frac{49}{93}\right)\) | \(e\left(\frac{203}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{431}{1116}\right)\) | \(e\left(\frac{34}{93}\right)\) | \(e\left(\frac{431}{558}\right)\) | \(e\left(\frac{301}{372}\right)\) | \(e\left(\frac{839}{1116}\right)\) | \(e\left(\frac{349}{558}\right)\) | \(e\left(\frac{59}{372}\right)\) | \(e\left(\frac{68}{93}\right)\) | \(e\left(\frac{109}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{1117}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{1116}\right)\) | \(e\left(\frac{89}{93}\right)\) | \(e\left(\frac{283}{558}\right)\) | \(e\left(\frac{353}{372}\right)\) | \(e\left(\frac{235}{1116}\right)\) | \(e\left(\frac{413}{558}\right)\) | \(e\left(\frac{283}{372}\right)\) | \(e\left(\frac{85}{93}\right)\) | \(e\left(\frac{113}{558}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{1117}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{1116}\right)\) | \(e\left(\frac{23}{93}\right)\) | \(e\left(\frac{163}{558}\right)\) | \(e\left(\frac{365}{372}\right)\) | \(e\left(\frac{439}{1116}\right)\) | \(e\left(\frac{299}{558}\right)\) | \(e\left(\frac{163}{372}\right)\) | \(e\left(\frac{46}{93}\right)\) | \(e\left(\frac{71}{558}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{1117}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{437}{1116}\right)\) | \(e\left(\frac{28}{93}\right)\) | \(e\left(\frac{437}{558}\right)\) | \(e\left(\frac{319}{372}\right)\) | \(e\left(\frac{773}{1116}\right)\) | \(e\left(\frac{271}{558}\right)\) | \(e\left(\frac{65}{372}\right)\) | \(e\left(\frac{56}{93}\right)\) | \(e\left(\frac{139}{558}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{1117}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{811}{1116}\right)\) | \(e\left(\frac{26}{93}\right)\) | \(e\left(\frac{253}{558}\right)\) | \(e\left(\frac{77}{372}\right)\) | \(e\left(\frac{7}{1116}\right)\) | \(e\left(\frac{245}{558}\right)\) | \(e\left(\frac{67}{372}\right)\) | \(e\left(\frac{52}{93}\right)\) | \(e\left(\frac{521}{558}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{1117}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{601}{1116}\right)\) | \(e\left(\frac{50}{93}\right)\) | \(e\left(\frac{43}{558}\right)\) | \(e\left(\frac{191}{372}\right)\) | \(e\left(\frac{85}{1116}\right)\) | \(e\left(\frac{185}{558}\right)\) | \(e\left(\frac{229}{372}\right)\) | \(e\left(\frac{7}{93}\right)\) | \(e\left(\frac{29}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{757}{1116}\right)\) | \(e\left(\frac{80}{93}\right)\) | \(e\left(\frac{199}{558}\right)\) | \(e\left(\frac{287}{372}\right)\) | \(e\left(\frac{601}{1116}\right)\) | \(e\left(\frac{389}{558}\right)\) | \(e\left(\frac{13}{372}\right)\) | \(e\left(\frac{67}{93}\right)\) | \(e\left(\frac{251}{558}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{1117}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{1116}\right)\) | \(e\left(\frac{46}{93}\right)\) | \(e\left(\frac{233}{558}\right)\) | \(e\left(\frac{79}{372}\right)\) | \(e\left(\frac{785}{1116}\right)\) | \(e\left(\frac{133}{558}\right)\) | \(e\left(\frac{233}{372}\right)\) | \(e\left(\frac{92}{93}\right)\) | \(e\left(\frac{235}{558}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{1117}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{851}{1116}\right)\) | \(e\left(\frac{79}{93}\right)\) | \(e\left(\frac{293}{558}\right)\) | \(e\left(\frac{73}{372}\right)\) | \(e\left(\frac{683}{1116}\right)\) | \(e\left(\frac{469}{558}\right)\) | \(e\left(\frac{107}{372}\right)\) | \(e\left(\frac{65}{93}\right)\) | \(e\left(\frac{535}{558}\right)\) | \(e\left(\frac{7}{12}\right)\) |