from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1283, base_ring=CyclotomicField(1282))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1283))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1283\) | |
| Conductor: | \(1283\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1282\) | 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: | $\Q(\zeta_{641})$ |
| Fixed field: | Number field defined by a degree 1282 polynomial (not computed) |
First 31 of 640 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1283}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1282}\right)\) | \(e\left(\frac{607}{641}\right)\) | \(e\left(\frac{1}{641}\right)\) | \(e\left(\frac{565}{1282}\right)\) | \(e\left(\frac{1215}{1282}\right)\) | \(e\left(\frac{1027}{1282}\right)\) | \(e\left(\frac{3}{1282}\right)\) | \(e\left(\frac{573}{641}\right)\) | \(e\left(\frac{283}{641}\right)\) | \(e\left(\frac{203}{641}\right)\) |
| \(\chi_{1283}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{565}{1282}\right)\) | \(e\left(\frac{20}{641}\right)\) | \(e\left(\frac{565}{641}\right)\) | \(e\left(\frac{7}{1282}\right)\) | \(e\left(\frac{605}{1282}\right)\) | \(e\left(\frac{791}{1282}\right)\) | \(e\left(\frac{413}{1282}\right)\) | \(e\left(\frac{40}{641}\right)\) | \(e\left(\frac{286}{641}\right)\) | \(e\left(\frac{597}{641}\right)\) |
| \(\chi_{1283}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1215}{1282}\right)\) | \(e\left(\frac{355}{641}\right)\) | \(e\left(\frac{574}{641}\right)\) | \(e\left(\frac{605}{1282}\right)\) | \(e\left(\frac{643}{1282}\right)\) | \(e\left(\frac{419}{1282}\right)\) | \(e\left(\frac{1081}{1282}\right)\) | \(e\left(\frac{69}{641}\right)\) | \(e\left(\frac{269}{641}\right)\) | \(e\left(\frac{501}{641}\right)\) |
| \(\chi_{1283}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1027}{1282}\right)\) | \(e\left(\frac{337}{641}\right)\) | \(e\left(\frac{386}{641}\right)\) | \(e\left(\frac{791}{1282}\right)\) | \(e\left(\frac{419}{1282}\right)\) | \(e\left(\frac{925}{1282}\right)\) | \(e\left(\frac{517}{1282}\right)\) | \(e\left(\frac{33}{641}\right)\) | \(e\left(\frac{268}{641}\right)\) | \(e\left(\frac{156}{641}\right)\) |
| \(\chi_{1283}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{1282}\right)\) | \(e\left(\frac{539}{641}\right)\) | \(e\left(\frac{3}{641}\right)\) | \(e\left(\frac{413}{1282}\right)\) | \(e\left(\frac{1081}{1282}\right)\) | \(e\left(\frac{517}{1282}\right)\) | \(e\left(\frac{9}{1282}\right)\) | \(e\left(\frac{437}{641}\right)\) | \(e\left(\frac{208}{641}\right)\) | \(e\left(\frac{609}{641}\right)\) |
| \(\chi_{1283}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{497}{1282}\right)\) | \(e\left(\frac{409}{641}\right)\) | \(e\left(\frac{497}{641}\right)\) | \(e\left(\frac{47}{1282}\right)\) | \(e\left(\frac{33}{1282}\right)\) | \(e\left(\frac{183}{1282}\right)\) | \(e\left(\frac{209}{1282}\right)\) | \(e\left(\frac{177}{641}\right)\) | \(e\left(\frac{272}{641}\right)\) | \(e\left(\frac{254}{641}\right)\) |
| \(\chi_{1283}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1147}{1282}\right)\) | \(e\left(\frac{103}{641}\right)\) | \(e\left(\frac{506}{641}\right)\) | \(e\left(\frac{645}{1282}\right)\) | \(e\left(\frac{71}{1282}\right)\) | \(e\left(\frac{1093}{1282}\right)\) | \(e\left(\frac{877}{1282}\right)\) | \(e\left(\frac{206}{641}\right)\) | \(e\left(\frac{255}{641}\right)\) | \(e\left(\frac{158}{641}\right)\) |
| \(\chi_{1283}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{567}{1282}\right)\) | \(e\left(\frac{593}{641}\right)\) | \(e\left(\frac{567}{641}\right)\) | \(e\left(\frac{1137}{1282}\right)\) | \(e\left(\frac{471}{1282}\right)\) | \(e\left(\frac{281}{1282}\right)\) | \(e\left(\frac{419}{1282}\right)\) | \(e\left(\frac{545}{641}\right)\) | \(e\left(\frac{211}{641}\right)\) | \(e\left(\frac{362}{641}\right)\) |
| \(\chi_{1283}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{959}{1282}\right)\) | \(e\left(\frac{85}{641}\right)\) | \(e\left(\frac{318}{641}\right)\) | \(e\left(\frac{831}{1282}\right)\) | \(e\left(\frac{1129}{1282}\right)\) | \(e\left(\frac{317}{1282}\right)\) | \(e\left(\frac{313}{1282}\right)\) | \(e\left(\frac{170}{641}\right)\) | \(e\left(\frac{254}{641}\right)\) | \(e\left(\frac{454}{641}\right)\) |
| \(\chi_{1283}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{407}{1282}\right)\) | \(e\left(\frac{264}{641}\right)\) | \(e\left(\frac{407}{641}\right)\) | \(e\left(\frac{477}{1282}\right)\) | \(e\left(\frac{935}{1282}\right)\) | \(e\left(\frac{57}{1282}\right)\) | \(e\left(\frac{1221}{1282}\right)\) | \(e\left(\frac{528}{641}\right)\) | \(e\left(\frac{442}{641}\right)\) | \(e\left(\frac{573}{641}\right)\) |
| \(\chi_{1283}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{817}{1282}\right)\) | \(e\left(\frac{426}{641}\right)\) | \(e\left(\frac{176}{641}\right)\) | \(e\left(\frac{85}{1282}\right)\) | \(e\left(\frac{387}{1282}\right)\) | \(e\left(\frac{631}{1282}\right)\) | \(e\left(\frac{1169}{1282}\right)\) | \(e\left(\frac{211}{641}\right)\) | \(e\left(\frac{451}{641}\right)\) | \(e\left(\frac{473}{641}\right)\) |
| \(\chi_{1283}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1217}{1282}\right)\) | \(e\left(\frac{287}{641}\right)\) | \(e\left(\frac{576}{641}\right)\) | \(e\left(\frac{453}{1282}\right)\) | \(e\left(\frac{509}{1282}\right)\) | \(e\left(\frac{1191}{1282}\right)\) | \(e\left(\frac{1087}{1282}\right)\) | \(e\left(\frac{574}{641}\right)\) | \(e\left(\frac{194}{641}\right)\) | \(e\left(\frac{266}{641}\right)\) |
| \(\chi_{1283}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1015}{1282}\right)\) | \(e\left(\frac{104}{641}\right)\) | \(e\left(\frac{374}{641}\right)\) | \(e\left(\frac{421}{1282}\right)\) | \(e\left(\frac{1223}{1282}\right)\) | \(e\left(\frac{139}{1282}\right)\) | \(e\left(\frac{481}{1282}\right)\) | \(e\left(\frac{208}{641}\right)\) | \(e\left(\frac{77}{641}\right)\) | \(e\left(\frac{284}{641}\right)\) |
| \(\chi_{1283}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1029}{1282}\right)\) | \(e\left(\frac{269}{641}\right)\) | \(e\left(\frac{388}{641}\right)\) | \(e\left(\frac{639}{1282}\right)\) | \(e\left(\frac{285}{1282}\right)\) | \(e\left(\frac{415}{1282}\right)\) | \(e\left(\frac{523}{1282}\right)\) | \(e\left(\frac{538}{641}\right)\) | \(e\left(\frac{193}{641}\right)\) | \(e\left(\frac{562}{641}\right)\) |
| \(\chi_{1283}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{1282}\right)\) | \(e\left(\frac{471}{641}\right)\) | \(e\left(\frac{5}{641}\right)\) | \(e\left(\frac{261}{1282}\right)\) | \(e\left(\frac{947}{1282}\right)\) | \(e\left(\frac{7}{1282}\right)\) | \(e\left(\frac{15}{1282}\right)\) | \(e\left(\frac{301}{641}\right)\) | \(e\left(\frac{133}{641}\right)\) | \(e\left(\frac{374}{641}\right)\) |
| \(\chi_{1283}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{865}{1282}\right)\) | \(e\left(\frac{76}{641}\right)\) | \(e\left(\frac{224}{641}\right)\) | \(e\left(\frac{283}{1282}\right)\) | \(e\left(\frac{1017}{1282}\right)\) | \(e\left(\frac{1211}{1282}\right)\) | \(e\left(\frac{31}{1282}\right)\) | \(e\left(\frac{152}{641}\right)\) | \(e\left(\frac{574}{641}\right)\) | \(e\left(\frac{602}{641}\right)\) |
| \(\chi_{1283}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{1282}\right)\) | \(e\left(\frac{61}{641}\right)\) | \(e\left(\frac{281}{641}\right)\) | \(e\left(\frac{1079}{1282}\right)\) | \(e\left(\frac{403}{1282}\right)\) | \(e\left(\frac{137}{1282}\right)\) | \(e\left(\frac{843}{1282}\right)\) | \(e\left(\frac{122}{641}\right)\) | \(e\left(\frac{39}{641}\right)\) | \(e\left(\frac{635}{641}\right)\) |
| \(\chi_{1283}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1255}{1282}\right)\) | \(e\left(\frac{277}{641}\right)\) | \(e\left(\frac{614}{641}\right)\) | \(e\left(\frac{129}{1282}\right)\) | \(e\left(\frac{527}{1282}\right)\) | \(e\left(\frac{475}{1282}\right)\) | \(e\left(\frac{1201}{1282}\right)\) | \(e\left(\frac{554}{641}\right)\) | \(e\left(\frac{51}{641}\right)\) | \(e\left(\frac{288}{641}\right)\) |
| \(\chi_{1283}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{529}{1282}\right)\) | \(e\left(\frac{603}{641}\right)\) | \(e\left(\frac{529}{641}\right)\) | \(e\left(\frac{179}{1282}\right)\) | \(e\left(\frac{453}{1282}\right)\) | \(e\left(\frac{997}{1282}\right)\) | \(e\left(\frac{305}{1282}\right)\) | \(e\left(\frac{565}{641}\right)\) | \(e\left(\frac{354}{641}\right)\) | \(e\left(\frac{340}{641}\right)\) |
| \(\chi_{1283}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{429}{1282}\right)\) | \(e\left(\frac{157}{641}\right)\) | \(e\left(\frac{429}{641}\right)\) | \(e\left(\frac{87}{1282}\right)\) | \(e\left(\frac{743}{1282}\right)\) | \(e\left(\frac{857}{1282}\right)\) | \(e\left(\frac{5}{1282}\right)\) | \(e\left(\frac{314}{641}\right)\) | \(e\left(\frac{258}{641}\right)\) | \(e\left(\frac{552}{641}\right)\) |
| \(\chi_{1283}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{591}{1282}\right)\) | \(e\left(\frac{418}{641}\right)\) | \(e\left(\frac{591}{641}\right)\) | \(e\left(\frac{595}{1282}\right)\) | \(e\left(\frac{145}{1282}\right)\) | \(e\left(\frac{571}{1282}\right)\) | \(e\left(\frac{491}{1282}\right)\) | \(e\left(\frac{195}{641}\right)\) | \(e\left(\frac{593}{641}\right)\) | \(e\left(\frac{106}{641}\right)\) |
| \(\chi_{1283}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1131}{1282}\right)\) | \(e\left(\frac{6}{641}\right)\) | \(e\left(\frac{490}{641}\right)\) | \(e\left(\frac{579}{1282}\right)\) | \(e\left(\frac{1143}{1282}\right)\) | \(e\left(\frac{45}{1282}\right)\) | \(e\left(\frac{829}{1282}\right)\) | \(e\left(\frac{12}{641}\right)\) | \(e\left(\frac{214}{641}\right)\) | \(e\left(\frac{115}{641}\right)\) |
| \(\chi_{1283}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1079}{1282}\right)\) | \(e\left(\frac{492}{641}\right)\) | \(e\left(\frac{438}{641}\right)\) | \(e\left(\frac{685}{1282}\right)\) | \(e\left(\frac{781}{1282}\right)\) | \(e\left(\frac{485}{1282}\right)\) | \(e\left(\frac{673}{1282}\right)\) | \(e\left(\frac{343}{641}\right)\) | \(e\left(\frac{241}{641}\right)\) | \(e\left(\frac{456}{641}\right)\) |
| \(\chi_{1283}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{971}{1282}\right)\) | \(e\left(\frac{318}{641}\right)\) | \(e\left(\frac{330}{641}\right)\) | \(e\left(\frac{1201}{1282}\right)\) | \(e\left(\frac{325}{1282}\right)\) | \(e\left(\frac{1103}{1282}\right)\) | \(e\left(\frac{349}{1282}\right)\) | \(e\left(\frac{636}{641}\right)\) | \(e\left(\frac{445}{641}\right)\) | \(e\left(\frac{326}{641}\right)\) |
| \(\chi_{1283}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1261}{1282}\right)\) | \(e\left(\frac{73}{641}\right)\) | \(e\left(\frac{620}{641}\right)\) | \(e\left(\frac{955}{1282}\right)\) | \(e\left(\frac{125}{1282}\right)\) | \(e\left(\frac{227}{1282}\right)\) | \(e\left(\frac{1219}{1282}\right)\) | \(e\left(\frac{146}{641}\right)\) | \(e\left(\frac{467}{641}\right)\) | \(e\left(\frac{224}{641}\right)\) |
| \(\chi_{1283}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{499}{1282}\right)\) | \(e\left(\frac{341}{641}\right)\) | \(e\left(\frac{499}{641}\right)\) | \(e\left(\frac{1177}{1282}\right)\) | \(e\left(\frac{1181}{1282}\right)\) | \(e\left(\frac{955}{1282}\right)\) | \(e\left(\frac{215}{1282}\right)\) | \(e\left(\frac{41}{641}\right)\) | \(e\left(\frac{197}{641}\right)\) | \(e\left(\frac{19}{641}\right)\) |
| \(\chi_{1283}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{965}{1282}\right)\) | \(e\left(\frac{522}{641}\right)\) | \(e\left(\frac{324}{641}\right)\) | \(e\left(\frac{375}{1282}\right)\) | \(e\left(\frac{727}{1282}\right)\) | \(e\left(\frac{69}{1282}\right)\) | \(e\left(\frac{331}{1282}\right)\) | \(e\left(\frac{403}{641}\right)\) | \(e\left(\frac{29}{641}\right)\) | \(e\left(\frac{390}{641}\right)\) |
| \(\chi_{1283}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{603}{1282}\right)\) | \(e\left(\frac{10}{641}\right)\) | \(e\left(\frac{603}{641}\right)\) | \(e\left(\frac{965}{1282}\right)\) | \(e\left(\frac{623}{1282}\right)\) | \(e\left(\frac{75}{1282}\right)\) | \(e\left(\frac{527}{1282}\right)\) | \(e\left(\frac{20}{641}\right)\) | \(e\left(\frac{143}{641}\right)\) | \(e\left(\frac{619}{641}\right)\) |
| \(\chi_{1283}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{891}{1282}\right)\) | \(e\left(\frac{474}{641}\right)\) | \(e\left(\frac{250}{641}\right)\) | \(e\left(\frac{871}{1282}\right)\) | \(e\left(\frac{557}{1282}\right)\) | \(e\left(\frac{991}{1282}\right)\) | \(e\left(\frac{109}{1282}\right)\) | \(e\left(\frac{307}{641}\right)\) | \(e\left(\frac{240}{641}\right)\) | \(e\left(\frac{111}{641}\right)\) |
| \(\chi_{1283}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{297}{1282}\right)\) | \(e\left(\frac{158}{641}\right)\) | \(e\left(\frac{297}{641}\right)\) | \(e\left(\frac{1145}{1282}\right)\) | \(e\left(\frac{613}{1282}\right)\) | \(e\left(\frac{1185}{1282}\right)\) | \(e\left(\frac{891}{1282}\right)\) | \(e\left(\frac{316}{641}\right)\) | \(e\left(\frac{80}{641}\right)\) | \(e\left(\frac{37}{641}\right)\) |
| \(\chi_{1283}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{339}{1282}\right)\) | \(e\left(\frac{12}{641}\right)\) | \(e\left(\frac{339}{641}\right)\) | \(e\left(\frac{517}{1282}\right)\) | \(e\left(\frac{363}{1282}\right)\) | \(e\left(\frac{731}{1282}\right)\) | \(e\left(\frac{1017}{1282}\right)\) | \(e\left(\frac{24}{641}\right)\) | \(e\left(\frac{428}{641}\right)\) | \(e\left(\frac{230}{641}\right)\) |