from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35937, base_ring=CyclotomicField(2178))
M = H._module
chi = DirichletCharacter(H, M([1331,522]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,35937))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(35937\) | |
| Conductor: | \(35937\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2178\) | 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: | $\Q(\zeta_{1089})$ |
| Fixed field: | Number field defined by a degree 2178 polynomial (not computed) |
First 29 of 660 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{35937}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1853}{2178}\right)\) | \(e\left(\frac{764}{1089}\right)\) | \(e\left(\frac{337}{2178}\right)\) | \(e\left(\frac{298}{1089}\right)\) | \(e\left(\frac{401}{726}\right)\) | \(e\left(\frac{2}{363}\right)\) | \(e\left(\frac{797}{1089}\right)\) | \(e\left(\frac{271}{2178}\right)\) | \(e\left(\frac{439}{1089}\right)\) | \(e\left(\frac{199}{726}\right)\) |
| \(\chi_{35937}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1129}{2178}\right)\) | \(e\left(\frac{40}{1089}\right)\) | \(e\left(\frac{947}{2178}\right)\) | \(e\left(\frac{734}{1089}\right)\) | \(e\left(\frac{403}{726}\right)\) | \(e\left(\frac{346}{363}\right)\) | \(e\left(\frac{304}{1089}\right)\) | \(e\left(\frac{419}{2178}\right)\) | \(e\left(\frac{80}{1089}\right)\) | \(e\left(\frac{305}{726}\right)\) |
| \(\chi_{35937}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{739}{2178}\right)\) | \(e\left(\frac{739}{1089}\right)\) | \(e\left(\frac{1787}{2178}\right)\) | \(e\left(\frac{656}{1089}\right)\) | \(e\left(\frac{13}{726}\right)\) | \(e\left(\frac{58}{363}\right)\) | \(e\left(\frac{607}{1089}\right)\) | \(e\left(\frac{2051}{2178}\right)\) | \(e\left(\frac{389}{1089}\right)\) | \(e\left(\frac{689}{726}\right)\) |
| \(\chi_{35937}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{875}{2178}\right)\) | \(e\left(\frac{875}{1089}\right)\) | \(e\left(\frac{433}{2178}\right)\) | \(e\left(\frac{538}{1089}\right)\) | \(e\left(\frac{149}{726}\right)\) | \(e\left(\frac{218}{363}\right)\) | \(e\left(\frac{116}{1089}\right)\) | \(e\left(\frac{1951}{2178}\right)\) | \(e\left(\frac{661}{1089}\right)\) | \(e\left(\frac{637}{726}\right)\) |
| \(\chi_{35937}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1933}{2178}\right)\) | \(e\left(\frac{844}{1089}\right)\) | \(e\left(\frac{53}{2178}\right)\) | \(e\left(\frac{677}{1089}\right)\) | \(e\left(\frac{481}{726}\right)\) | \(e\left(\frac{331}{363}\right)\) | \(e\left(\frac{316}{1089}\right)\) | \(e\left(\frac{1109}{2178}\right)\) | \(e\left(\frac{599}{1089}\right)\) | \(e\left(\frac{83}{726}\right)\) |
| \(\chi_{35937}(320,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1673}{2178}\right)\) | \(e\left(\frac{584}{1089}\right)\) | \(e\left(\frac{2065}{2178}\right)\) | \(e\left(\frac{262}{1089}\right)\) | \(e\left(\frac{221}{726}\right)\) | \(e\left(\frac{260}{363}\right)\) | \(e\left(\frac{518}{1089}\right)\) | \(e\left(\frac{19}{2178}\right)\) | \(e\left(\frac{79}{1089}\right)\) | \(e\left(\frac{97}{726}\right)\) |
| \(\chi_{35937}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{355}{2178}\right)\) | \(e\left(\frac{355}{1089}\right)\) | \(e\left(\frac{101}{2178}\right)\) | \(e\left(\frac{797}{1089}\right)\) | \(e\left(\frac{355}{726}\right)\) | \(e\left(\frac{76}{363}\right)\) | \(e\left(\frac{520}{1089}\right)\) | \(e\left(\frac{1949}{2178}\right)\) | \(e\left(\frac{710}{1089}\right)\) | \(e\left(\frac{665}{726}\right)\) |
| \(\chi_{35937}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1877}{2178}\right)\) | \(e\left(\frac{788}{1089}\right)\) | \(e\left(\frac{1123}{2178}\right)\) | \(e\left(\frac{85}{1089}\right)\) | \(e\left(\frac{425}{726}\right)\) | \(e\left(\frac{137}{363}\right)\) | \(e\left(\frac{326}{1089}\right)\) | \(e\left(\frac{2047}{2178}\right)\) | \(e\left(\frac{487}{1089}\right)\) | \(e\left(\frac{19}{726}\right)\) |
| \(\chi_{35937}(452,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{2178}\right)\) | \(e\left(\frac{361}{1089}\right)\) | \(e\left(\frac{1931}{2178}\right)\) | \(e\left(\frac{1016}{1089}\right)\) | \(e\left(\frac{361}{726}\right)\) | \(e\left(\frac{19}{363}\right)\) | \(e\left(\frac{130}{1089}\right)\) | \(e\left(\frac{215}{2178}\right)\) | \(e\left(\frac{722}{1089}\right)\) | \(e\left(\frac{257}{726}\right)\) |
| \(\chi_{35937}(518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1487}{2178}\right)\) | \(e\left(\frac{398}{1089}\right)\) | \(e\left(\frac{1963}{2178}\right)\) | \(e\left(\frac{7}{1089}\right)\) | \(e\left(\frac{35}{726}\right)\) | \(e\left(\frac{212}{363}\right)\) | \(e\left(\frac{629}{1089}\right)\) | \(e\left(\frac{1501}{2178}\right)\) | \(e\left(\frac{796}{1089}\right)\) | \(e\left(\frac{403}{726}\right)\) |
| \(\chi_{35937}(551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1951}{2178}\right)\) | \(e\left(\frac{862}{1089}\right)\) | \(e\left(\frac{1187}{2178}\right)\) | \(e\left(\frac{245}{1089}\right)\) | \(e\left(\frac{499}{726}\right)\) | \(e\left(\frac{160}{363}\right)\) | \(e\left(\frac{235}{1089}\right)\) | \(e\left(\frac{263}{2178}\right)\) | \(e\left(\frac{635}{1089}\right)\) | \(e\left(\frac{311}{726}\right)\) |
| \(\chi_{35937}(617,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{2178}\right)\) | \(e\left(\frac{503}{1089}\right)\) | \(e\left(\frac{229}{2178}\right)\) | \(e\left(\frac{28}{1089}\right)\) | \(e\left(\frac{503}{726}\right)\) | \(e\left(\frac{122}{363}\right)\) | \(e\left(\frac{338}{1089}\right)\) | \(e\left(\frac{559}{2178}\right)\) | \(e\left(\frac{1006}{1089}\right)\) | \(e\left(\frac{523}{726}\right)\) |
| \(\chi_{35937}(650,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{769}{2178}\right)\) | \(e\left(\frac{769}{1089}\right)\) | \(e\left(\frac{47}{2178}\right)\) | \(e\left(\frac{662}{1089}\right)\) | \(e\left(\frac{43}{726}\right)\) | \(e\left(\frac{136}{363}\right)\) | \(e\left(\frac{835}{1089}\right)\) | \(e\left(\frac{2093}{2178}\right)\) | \(e\left(\frac{449}{1089}\right)\) | \(e\left(\frac{101}{726}\right)\) |
| \(\chi_{35937}(716,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1103}{2178}\right)\) | \(e\left(\frac{14}{1089}\right)\) | \(e\left(\frac{277}{2178}\right)\) | \(e\left(\frac{148}{1089}\right)\) | \(e\left(\frac{377}{726}\right)\) | \(e\left(\frac{230}{363}\right)\) | \(e\left(\frac{542}{1089}\right)\) | \(e\left(\frac{1399}{2178}\right)\) | \(e\left(\frac{28}{1089}\right)\) | \(e\left(\frac{379}{726}\right)\) |
| \(\chi_{35937}(749,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1171}{2178}\right)\) | \(e\left(\frac{82}{1089}\right)\) | \(e\left(\frac{689}{2178}\right)\) | \(e\left(\frac{89}{1089}\right)\) | \(e\left(\frac{445}{726}\right)\) | \(e\left(\frac{310}{363}\right)\) | \(e\left(\frac{841}{1089}\right)\) | \(e\left(\frac{1349}{2178}\right)\) | \(e\left(\frac{164}{1089}\right)\) | \(e\left(\frac{353}{726}\right)\) |
| \(\chi_{35937}(815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1109}{2178}\right)\) | \(e\left(\frac{20}{1089}\right)\) | \(e\left(\frac{2107}{2178}\right)\) | \(e\left(\frac{367}{1089}\right)\) | \(e\left(\frac{383}{726}\right)\) | \(e\left(\frac{173}{363}\right)\) | \(e\left(\frac{152}{1089}\right)\) | \(e\left(\frac{1843}{2178}\right)\) | \(e\left(\frac{40}{1089}\right)\) | \(e\left(\frac{697}{726}\right)\) |
| \(\chi_{35937}(914,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{521}{2178}\right)\) | \(e\left(\frac{521}{1089}\right)\) | \(e\left(\frac{1363}{2178}\right)\) | \(e\left(\frac{685}{1089}\right)\) | \(e\left(\frac{521}{726}\right)\) | \(e\left(\frac{314}{363}\right)\) | \(e\left(\frac{257}{1089}\right)\) | \(e\left(\frac{1891}{2178}\right)\) | \(e\left(\frac{1042}{1089}\right)\) | \(e\left(\frac{25}{726}\right)\) |
| \(\chi_{35937}(947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{2178}\right)\) | \(e\left(\frac{193}{1089}\right)\) | \(e\left(\frac{785}{2178}\right)\) | \(e\left(\frac{329}{1089}\right)\) | \(e\left(\frac{193}{726}\right)\) | \(e\left(\frac{163}{363}\right)\) | \(e\left(\frac{160}{1089}\right)\) | \(e\left(\frac{851}{2178}\right)\) | \(e\left(\frac{386}{1089}\right)\) | \(e\left(\frac{65}{726}\right)\) |
| \(\chi_{35937}(1013,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1517}{2178}\right)\) | \(e\left(\frac{428}{1089}\right)\) | \(e\left(\frac{223}{2178}\right)\) | \(e\left(\frac{13}{1089}\right)\) | \(e\left(\frac{65}{726}\right)\) | \(e\left(\frac{290}{363}\right)\) | \(e\left(\frac{857}{1089}\right)\) | \(e\left(\frac{1543}{2178}\right)\) | \(e\left(\frac{856}{1089}\right)\) | \(e\left(\frac{541}{726}\right)\) |
| \(\chi_{35937}(1046,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{991}{2178}\right)\) | \(e\left(\frac{991}{1089}\right)\) | \(e\left(\frac{239}{2178}\right)\) | \(e\left(\frac{53}{1089}\right)\) | \(e\left(\frac{265}{726}\right)\) | \(e\left(\frac{205}{363}\right)\) | \(e\left(\frac{562}{1089}\right)\) | \(e\left(\frac{1097}{2178}\right)\) | \(e\left(\frac{893}{1089}\right)\) | \(e\left(\frac{251}{726}\right)\) |
| \(\chi_{35937}(1112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1919}{2178}\right)\) | \(e\left(\frac{830}{1089}\right)\) | \(e\left(\frac{865}{2178}\right)\) | \(e\left(\frac{529}{1089}\right)\) | \(e\left(\frac{467}{726}\right)\) | \(e\left(\frac{101}{363}\right)\) | \(e\left(\frac{863}{1089}\right)\) | \(e\left(\frac{799}{2178}\right)\) | \(e\left(\frac{571}{1089}\right)\) | \(e\left(\frac{67}{726}\right)\) |
| \(\chi_{35937}(1145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1195}{2178}\right)\) | \(e\left(\frac{106}{1089}\right)\) | \(e\left(\frac{1475}{2178}\right)\) | \(e\left(\frac{965}{1089}\right)\) | \(e\left(\frac{469}{726}\right)\) | \(e\left(\frac{82}{363}\right)\) | \(e\left(\frac{370}{1089}\right)\) | \(e\left(\frac{947}{2178}\right)\) | \(e\left(\frac{212}{1089}\right)\) | \(e\left(\frac{173}{726}\right)\) |
| \(\chi_{35937}(1244,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{805}{2178}\right)\) | \(e\left(\frac{805}{1089}\right)\) | \(e\left(\frac{137}{2178}\right)\) | \(e\left(\frac{887}{1089}\right)\) | \(e\left(\frac{79}{726}\right)\) | \(e\left(\frac{157}{363}\right)\) | \(e\left(\frac{673}{1089}\right)\) | \(e\left(\frac{401}{2178}\right)\) | \(e\left(\frac{521}{1089}\right)\) | \(e\left(\frac{557}{726}\right)\) |
| \(\chi_{35937}(1310,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{941}{2178}\right)\) | \(e\left(\frac{941}{1089}\right)\) | \(e\left(\frac{961}{2178}\right)\) | \(e\left(\frac{769}{1089}\right)\) | \(e\left(\frac{215}{726}\right)\) | \(e\left(\frac{317}{363}\right)\) | \(e\left(\frac{182}{1089}\right)\) | \(e\left(\frac{301}{2178}\right)\) | \(e\left(\frac{793}{1089}\right)\) | \(e\left(\frac{505}{726}\right)\) |
| \(\chi_{35937}(1343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1999}{2178}\right)\) | \(e\left(\frac{910}{1089}\right)\) | \(e\left(\frac{581}{2178}\right)\) | \(e\left(\frac{908}{1089}\right)\) | \(e\left(\frac{547}{726}\right)\) | \(e\left(\frac{67}{363}\right)\) | \(e\left(\frac{382}{1089}\right)\) | \(e\left(\frac{1637}{2178}\right)\) | \(e\left(\frac{731}{1089}\right)\) | \(e\left(\frac{677}{726}\right)\) |
| \(\chi_{35937}(1409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1739}{2178}\right)\) | \(e\left(\frac{650}{1089}\right)\) | \(e\left(\frac{415}{2178}\right)\) | \(e\left(\frac{493}{1089}\right)\) | \(e\left(\frac{287}{726}\right)\) | \(e\left(\frac{359}{363}\right)\) | \(e\left(\frac{584}{1089}\right)\) | \(e\left(\frac{547}{2178}\right)\) | \(e\left(\frac{211}{1089}\right)\) | \(e\left(\frac{691}{726}\right)\) |
| \(\chi_{35937}(1442,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{2178}\right)\) | \(e\left(\frac{421}{1089}\right)\) | \(e\left(\frac{629}{2178}\right)\) | \(e\left(\frac{1028}{1089}\right)\) | \(e\left(\frac{421}{726}\right)\) | \(e\left(\frac{175}{363}\right)\) | \(e\left(\frac{586}{1089}\right)\) | \(e\left(\frac{299}{2178}\right)\) | \(e\left(\frac{842}{1089}\right)\) | \(e\left(\frac{533}{726}\right)\) |
| \(\chi_{35937}(1508,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1943}{2178}\right)\) | \(e\left(\frac{854}{1089}\right)\) | \(e\left(\frac{1651}{2178}\right)\) | \(e\left(\frac{316}{1089}\right)\) | \(e\left(\frac{491}{726}\right)\) | \(e\left(\frac{236}{363}\right)\) | \(e\left(\frac{392}{1089}\right)\) | \(e\left(\frac{397}{2178}\right)\) | \(e\left(\frac{619}{1089}\right)\) | \(e\left(\frac{613}{726}\right)\) |
| \(\chi_{35937}(1541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{427}{2178}\right)\) | \(e\left(\frac{427}{1089}\right)\) | \(e\left(\frac{281}{2178}\right)\) | \(e\left(\frac{158}{1089}\right)\) | \(e\left(\frac{427}{726}\right)\) | \(e\left(\frac{118}{363}\right)\) | \(e\left(\frac{196}{1089}\right)\) | \(e\left(\frac{743}{2178}\right)\) | \(e\left(\frac{854}{1089}\right)\) | \(e\left(\frac{125}{726}\right)\) |