from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4027, base_ring=CyclotomicField(1342))
M = H._module
chi = DirichletCharacter(H, M([125]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,4027))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4027\) | |
| Conductor: | \(4027\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1342\) | 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: | $\Q(\zeta_{671})$ |
| Fixed field: | Number field defined by a degree 1342 polynomial (not computed) |
First 31 of 600 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4027}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1247}{1342}\right)\) | \(e\left(\frac{125}{1342}\right)\) | \(e\left(\frac{576}{671}\right)\) | \(e\left(\frac{507}{1342}\right)\) | \(e\left(\frac{15}{671}\right)\) | \(e\left(\frac{827}{1342}\right)\) | \(e\left(\frac{1057}{1342}\right)\) | \(e\left(\frac{125}{671}\right)\) | \(e\left(\frac{206}{671}\right)\) | \(e\left(\frac{1255}{1342}\right)\) |
| \(\chi_{4027}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{827}{1342}\right)\) | \(e\left(\frac{607}{1342}\right)\) | \(e\left(\frac{156}{671}\right)\) | \(e\left(\frac{347}{1342}\right)\) | \(e\left(\frac{46}{671}\right)\) | \(e\left(\frac{881}{1342}\right)\) | \(e\left(\frac{1139}{1342}\right)\) | \(e\left(\frac{607}{671}\right)\) | \(e\left(\frac{587}{671}\right)\) | \(e\left(\frac{941}{1342}\right)\) |
| \(\chi_{4027}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1057}{1342}\right)\) | \(e\left(\frac{375}{1342}\right)\) | \(e\left(\frac{386}{671}\right)\) | \(e\left(\frac{179}{1342}\right)\) | \(e\left(\frac{45}{671}\right)\) | \(e\left(\frac{1139}{1342}\right)\) | \(e\left(\frac{487}{1342}\right)\) | \(e\left(\frac{375}{671}\right)\) | \(e\left(\frac{618}{671}\right)\) | \(e\left(\frac{1081}{1342}\right)\) |
| \(\chi_{4027}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{1342}\right)\) | \(e\left(\frac{147}{1342}\right)\) | \(e\left(\frac{103}{671}\right)\) | \(e\left(\frac{199}{1342}\right)\) | \(e\left(\frac{125}{671}\right)\) | \(e\left(\frac{629}{1342}\right)\) | \(e\left(\frac{309}{1342}\right)\) | \(e\left(\frac{147}{671}\right)\) | \(e\left(\frac{151}{671}\right)\) | \(e\left(\frac{617}{1342}\right)\) |
| \(\chi_{4027}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{375}{1342}\right)\) | \(e\left(\frac{1}{1342}\right)\) | \(e\left(\frac{375}{671}\right)\) | \(e\left(\frac{47}{1342}\right)\) | \(e\left(\frac{188}{671}\right)\) | \(e\left(\frac{479}{1342}\right)\) | \(e\left(\frac{1125}{1342}\right)\) | \(e\left(\frac{1}{671}\right)\) | \(e\left(\frac{211}{671}\right)\) | \(e\left(\frac{1191}{1342}\right)\) |
| \(\chi_{4027}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{637}{1342}\right)\) | \(e\left(\frac{857}{1342}\right)\) | \(e\left(\frac{637}{671}\right)\) | \(e\left(\frac{19}{1342}\right)\) | \(e\left(\frac{76}{671}\right)\) | \(e\left(\frac{1193}{1342}\right)\) | \(e\left(\frac{569}{1342}\right)\) | \(e\left(\frac{186}{671}\right)\) | \(e\left(\frac{328}{671}\right)\) | \(e\left(\frac{767}{1342}\right)\) |
| \(\chi_{4027}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{537}{1342}\right)\) | \(e\left(\frac{141}{1342}\right)\) | \(e\left(\frac{537}{671}\right)\) | \(e\left(\frac{1259}{1342}\right)\) | \(e\left(\frac{339}{671}\right)\) | \(e\left(\frac{439}{1342}\right)\) | \(e\left(\frac{269}{1342}\right)\) | \(e\left(\frac{141}{671}\right)\) | \(e\left(\frac{227}{671}\right)\) | \(e\left(\frac{181}{1342}\right)\) |
| \(\chi_{4027}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{867}{1342}\right)\) | \(e\left(\frac{625}{1342}\right)\) | \(e\left(\frac{196}{671}\right)\) | \(e\left(\frac{1193}{1342}\right)\) | \(e\left(\frac{75}{671}\right)\) | \(e\left(\frac{109}{1342}\right)\) | \(e\left(\frac{1259}{1342}\right)\) | \(e\left(\frac{625}{671}\right)\) | \(e\left(\frac{359}{671}\right)\) | \(e\left(\frac{907}{1342}\right)\) |
| \(\chi_{4027}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{1342}\right)\) | \(e\left(\frac{885}{1342}\right)\) | \(e\left(\frac{401}{671}\right)\) | \(e\left(\frac{1335}{1342}\right)\) | \(e\left(\frac{643}{671}\right)\) | \(e\left(\frac{1185}{1342}\right)\) | \(e\left(\frac{1203}{1342}\right)\) | \(e\left(\frac{214}{671}\right)\) | \(e\left(\frac{197}{671}\right)\) | \(e\left(\frac{565}{1342}\right)\) |
| \(\chi_{4027}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{775}{1342}\right)\) | \(e\left(\frac{181}{1342}\right)\) | \(e\left(\frac{104}{671}\right)\) | \(e\left(\frac{455}{1342}\right)\) | \(e\left(\frac{478}{671}\right)\) | \(e\left(\frac{811}{1342}\right)\) | \(e\left(\frac{983}{1342}\right)\) | \(e\left(\frac{181}{671}\right)\) | \(e\left(\frac{615}{671}\right)\) | \(e\left(\frac{851}{1342}\right)\) |
| \(\chi_{4027}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1285}{1342}\right)\) | \(e\left(\frac{75}{1342}\right)\) | \(e\left(\frac{614}{671}\right)\) | \(e\left(\frac{841}{1342}\right)\) | \(e\left(\frac{9}{671}\right)\) | \(e\left(\frac{1033}{1342}\right)\) | \(e\left(\frac{1171}{1342}\right)\) | \(e\left(\frac{75}{671}\right)\) | \(e\left(\frac{392}{671}\right)\) | \(e\left(\frac{753}{1342}\right)\) |
| \(\chi_{4027}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1137}{1342}\right)\) | \(e\left(\frac{411}{1342}\right)\) | \(e\left(\frac{466}{671}\right)\) | \(e\left(\frac{529}{1342}\right)\) | \(e\left(\frac{103}{671}\right)\) | \(e\left(\frac{937}{1342}\right)\) | \(e\left(\frac{727}{1342}\right)\) | \(e\left(\frac{411}{671}\right)\) | \(e\left(\frac{162}{671}\right)\) | \(e\left(\frac{1013}{1342}\right)\) |
| \(\chi_{4027}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1025}{1342}\right)\) | \(e\left(\frac{629}{1342}\right)\) | \(e\left(\frac{354}{671}\right)\) | \(e\left(\frac{39}{1342}\right)\) | \(e\left(\frac{156}{671}\right)\) | \(e\left(\frac{683}{1342}\right)\) | \(e\left(\frac{391}{1342}\right)\) | \(e\left(\frac{629}{671}\right)\) | \(e\left(\frac{532}{671}\right)\) | \(e\left(\frac{303}{1342}\right)\) |
| \(\chi_{4027}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{1342}\right)\) | \(e\left(\frac{1339}{1342}\right)\) | \(e\left(\frac{217}{671}\right)\) | \(e\left(\frac{1201}{1342}\right)\) | \(e\left(\frac{107}{671}\right)\) | \(e\left(\frac{1247}{1342}\right)\) | \(e\left(\frac{651}{1342}\right)\) | \(e\left(\frac{668}{671}\right)\) | \(e\left(\frac{38}{671}\right)\) | \(e\left(\frac{453}{1342}\right)\) |
| \(\chi_{4027}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1255}{1342}\right)\) | \(e\left(\frac{397}{1342}\right)\) | \(e\left(\frac{584}{671}\right)\) | \(e\left(\frac{1213}{1342}\right)\) | \(e\left(\frac{155}{671}\right)\) | \(e\left(\frac{941}{1342}\right)\) | \(e\left(\frac{1081}{1342}\right)\) | \(e\left(\frac{397}{671}\right)\) | \(e\left(\frac{563}{671}\right)\) | \(e\left(\frac{443}{1342}\right)\) |
| \(\chi_{4027}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{1342}\right)\) | \(e\left(\frac{623}{1342}\right)\) | \(e\left(\frac{117}{671}\right)\) | \(e\left(\frac{1099}{1342}\right)\) | \(e\left(\frac{370}{671}\right)\) | \(e\left(\frac{493}{1342}\right)\) | \(e\left(\frac{351}{1342}\right)\) | \(e\left(\frac{623}{671}\right)\) | \(e\left(\frac{608}{671}\right)\) | \(e\left(\frac{1209}{1342}\right)\) |
| \(\chi_{4027}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{1342}\right)\) | \(e\left(\frac{251}{1342}\right)\) | \(e\left(\frac{185}{671}\right)\) | \(e\left(\frac{1061}{1342}\right)\) | \(e\left(\frac{218}{671}\right)\) | \(e\left(\frac{791}{1342}\right)\) | \(e\left(\frac{555}{1342}\right)\) | \(e\left(\frac{251}{671}\right)\) | \(e\left(\frac{623}{671}\right)\) | \(e\left(\frac{1017}{1342}\right)\) |
| \(\chi_{4027}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{447}{1342}\right)\) | \(e\left(\frac{1107}{1342}\right)\) | \(e\left(\frac{447}{671}\right)\) | \(e\left(\frac{1033}{1342}\right)\) | \(e\left(\frac{106}{671}\right)\) | \(e\left(\frac{163}{1342}\right)\) | \(e\left(\frac{1341}{1342}\right)\) | \(e\left(\frac{436}{671}\right)\) | \(e\left(\frac{69}{671}\right)\) | \(e\left(\frac{593}{1342}\right)\) |
| \(\chi_{4027}(120,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{1342}\right)\) | \(e\left(\frac{391}{1342}\right)\) | \(e\left(\frac{347}{671}\right)\) | \(e\left(\frac{931}{1342}\right)\) | \(e\left(\frac{369}{671}\right)\) | \(e\left(\frac{751}{1342}\right)\) | \(e\left(\frac{1041}{1342}\right)\) | \(e\left(\frac{391}{671}\right)\) | \(e\left(\frac{639}{671}\right)\) | \(e\left(\frac{7}{1342}\right)\) |
| \(\chi_{4027}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{633}{1342}\right)\) | \(e\left(\frac{721}{1342}\right)\) | \(e\left(\frac{633}{671}\right)\) | \(e\left(\frac{337}{1342}\right)\) | \(e\left(\frac{6}{671}\right)\) | \(e\left(\frac{465}{1342}\right)\) | \(e\left(\frac{557}{1342}\right)\) | \(e\left(\frac{50}{671}\right)\) | \(e\left(\frac{485}{671}\right)\) | \(e\left(\frac{1173}{1342}\right)\) |
| \(\chi_{4027}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{829}{1342}\right)\) | \(e\left(\frac{675}{1342}\right)\) | \(e\left(\frac{158}{671}\right)\) | \(e\left(\frac{859}{1342}\right)\) | \(e\left(\frac{81}{671}\right)\) | \(e\left(\frac{1245}{1342}\right)\) | \(e\left(\frac{1145}{1342}\right)\) | \(e\left(\frac{4}{671}\right)\) | \(e\left(\frac{173}{671}\right)\) | \(e\left(\frac{67}{1342}\right)\) |
| \(\chi_{4027}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{1342}\right)\) | \(e\left(\frac{47}{1342}\right)\) | \(e\left(\frac{179}{671}\right)\) | \(e\left(\frac{867}{1342}\right)\) | \(e\left(\frac{113}{671}\right)\) | \(e\left(\frac{1041}{1342}\right)\) | \(e\left(\frac{537}{1342}\right)\) | \(e\left(\frac{47}{671}\right)\) | \(e\left(\frac{523}{671}\right)\) | \(e\left(\frac{955}{1342}\right)\) |
| \(\chi_{4027}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{677}{1342}\right)\) | \(e\left(\frac{875}{1342}\right)\) | \(e\left(\frac{6}{671}\right)\) | \(e\left(\frac{865}{1342}\right)\) | \(e\left(\frac{105}{671}\right)\) | \(e\left(\frac{421}{1342}\right)\) | \(e\left(\frac{689}{1342}\right)\) | \(e\left(\frac{204}{671}\right)\) | \(e\left(\frac{100}{671}\right)\) | \(e\left(\frac{733}{1342}\right)\) |
| \(\chi_{4027}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{961}{1342}\right)\) | \(e\left(\frac{1137}{1342}\right)\) | \(e\left(\frac{290}{671}\right)\) | \(e\left(\frac{1101}{1342}\right)\) | \(e\left(\frac{378}{671}\right)\) | \(e\left(\frac{1113}{1342}\right)\) | \(e\left(\frac{199}{1342}\right)\) | \(e\left(\frac{466}{671}\right)\) | \(e\left(\frac{360}{671}\right)\) | \(e\left(\frac{89}{1342}\right)\) |
| \(\chi_{4027}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{1342}\right)\) | \(e\left(\frac{283}{1342}\right)\) | \(e\left(\frac{107}{671}\right)\) | \(e\left(\frac{1223}{1342}\right)\) | \(e\left(\frac{195}{671}\right)\) | \(e\left(\frac{15}{1342}\right)\) | \(e\left(\frac{321}{1342}\right)\) | \(e\left(\frac{283}{671}\right)\) | \(e\left(\frac{665}{671}\right)\) | \(e\left(\frac{211}{1342}\right)\) |
| \(\chi_{4027}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{743}{1342}\right)\) | \(e\left(\frac{435}{1342}\right)\) | \(e\left(\frac{72}{671}\right)\) | \(e\left(\frac{315}{1342}\right)\) | \(e\left(\frac{589}{671}\right)\) | \(e\left(\frac{355}{1342}\right)\) | \(e\left(\frac{887}{1342}\right)\) | \(e\left(\frac{435}{671}\right)\) | \(e\left(\frac{529}{671}\right)\) | \(e\left(\frac{73}{1342}\right)\) |
| \(\chi_{4027}(146,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{1342}\right)\) | \(e\left(\frac{1081}{1342}\right)\) | \(e\left(\frac{91}{671}\right)\) | \(e\left(\frac{1153}{1342}\right)\) | \(e\left(\frac{586}{671}\right)\) | \(e\left(\frac{1129}{1342}\right)\) | \(e\left(\frac{273}{1342}\right)\) | \(e\left(\frac{410}{671}\right)\) | \(e\left(\frac{622}{671}\right)\) | \(e\left(\frac{493}{1342}\right)\) |
| \(\chi_{4027}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{1342}\right)\) | \(e\left(\frac{469}{1342}\right)\) | \(e\left(\frac{73}{671}\right)\) | \(e\left(\frac{571}{1342}\right)\) | \(e\left(\frac{271}{671}\right)\) | \(e\left(\frac{537}{1342}\right)\) | \(e\left(\frac{219}{1342}\right)\) | \(e\left(\frac{469}{671}\right)\) | \(e\left(\frac{322}{671}\right)\) | \(e\left(\frac{307}{1342}\right)\) |
| \(\chi_{4027}(186,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{887}{1342}\right)\) | \(e\left(\frac{1305}{1342}\right)\) | \(e\left(\frac{216}{671}\right)\) | \(e\left(\frac{945}{1342}\right)\) | \(e\left(\frac{425}{671}\right)\) | \(e\left(\frac{1065}{1342}\right)\) | \(e\left(\frac{1319}{1342}\right)\) | \(e\left(\frac{634}{671}\right)\) | \(e\left(\frac{245}{671}\right)\) | \(e\left(\frac{219}{1342}\right)\) |
| \(\chi_{4027}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{543}{1342}\right)\) | \(e\left(\frac{345}{1342}\right)\) | \(e\left(\frac{543}{671}\right)\) | \(e\left(\frac{111}{1342}\right)\) | \(e\left(\frac{444}{671}\right)\) | \(e\left(\frac{189}{1342}\right)\) | \(e\left(\frac{287}{1342}\right)\) | \(e\left(\frac{345}{671}\right)\) | \(e\left(\frac{327}{671}\right)\) | \(e\left(\frac{243}{1342}\right)\) |
| \(\chi_{4027}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{1342}\right)\) | \(e\left(\frac{1135}{1342}\right)\) | \(e\left(\frac{211}{671}\right)\) | \(e\left(\frac{1007}{1342}\right)\) | \(e\left(\frac{2}{671}\right)\) | \(e\left(\frac{155}{1342}\right)\) | \(e\left(\frac{633}{1342}\right)\) | \(e\left(\frac{464}{671}\right)\) | \(e\left(\frac{609}{671}\right)\) | \(e\left(\frac{391}{1342}\right)\) |