from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1279, base_ring=CyclotomicField(1278))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1279))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1279\) | |
| Conductor: | \(1279\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1278\) | 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: | $\Q(\zeta_{639})$ |
| Fixed field: | Number field defined by a degree 1278 polynomial (not computed) |
First 31 of 420 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1279}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{639}\right)\) | \(e\left(\frac{1}{1278}\right)\) | \(e\left(\frac{46}{639}\right)\) | \(e\left(\frac{455}{639}\right)\) | \(e\left(\frac{47}{1278}\right)\) | \(e\left(\frac{356}{639}\right)\) | \(e\left(\frac{23}{213}\right)\) | \(e\left(\frac{1}{639}\right)\) | \(e\left(\frac{478}{639}\right)\) | \(e\left(\frac{19}{426}\right)\) |
| \(\chi_{1279}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{442}{639}\right)\) | \(e\left(\frac{47}{1278}\right)\) | \(e\left(\frac{245}{639}\right)\) | \(e\left(\frac{298}{639}\right)\) | \(e\left(\frac{931}{1278}\right)\) | \(e\left(\frac{118}{639}\right)\) | \(e\left(\frac{16}{213}\right)\) | \(e\left(\frac{47}{639}\right)\) | \(e\left(\frac{101}{639}\right)\) | \(e\left(\frac{41}{426}\right)\) |
| \(\chi_{1279}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{539}{639}\right)\) | \(e\left(\frac{79}{1278}\right)\) | \(e\left(\frac{439}{639}\right)\) | \(e\left(\frac{161}{639}\right)\) | \(e\left(\frac{1157}{1278}\right)\) | \(e\left(\frac{8}{639}\right)\) | \(e\left(\frac{113}{213}\right)\) | \(e\left(\frac{79}{639}\right)\) | \(e\left(\frac{61}{639}\right)\) | \(e\left(\frac{223}{426}\right)\) |
| \(\chi_{1279}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{505}{639}\right)\) | \(e\left(\frac{911}{1278}\right)\) | \(e\left(\frac{371}{639}\right)\) | \(e\left(\frac{433}{639}\right)\) | \(e\left(\frac{643}{1278}\right)\) | \(e\left(\frac{343}{639}\right)\) | \(e\left(\frac{79}{213}\right)\) | \(e\left(\frac{272}{639}\right)\) | \(e\left(\frac{299}{639}\right)\) | \(e\left(\frac{269}{426}\right)\) |
| \(\chi_{1279}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{424}{639}\right)\) | \(e\left(\frac{713}{1278}\right)\) | \(e\left(\frac{209}{639}\right)\) | \(e\left(\frac{442}{639}\right)\) | \(e\left(\frac{283}{1278}\right)\) | \(e\left(\frac{145}{639}\right)\) | \(e\left(\frac{211}{213}\right)\) | \(e\left(\frac{74}{639}\right)\) | \(e\left(\frac{227}{639}\right)\) | \(e\left(\frac{341}{426}\right)\) |
| \(\chi_{1279}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{452}{639}\right)\) | \(e\left(\frac{103}{1278}\right)\) | \(e\left(\frac{265}{639}\right)\) | \(e\left(\frac{218}{639}\right)\) | \(e\left(\frac{1007}{1278}\right)\) | \(e\left(\frac{245}{639}\right)\) | \(e\left(\frac{26}{213}\right)\) | \(e\left(\frac{103}{639}\right)\) | \(e\left(\frac{31}{639}\right)\) | \(e\left(\frac{253}{426}\right)\) |
| \(\chi_{1279}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{639}\right)\) | \(e\left(\frac{139}{1278}\right)\) | \(e\left(\frac{4}{639}\right)\) | \(e\left(\frac{623}{639}\right)\) | \(e\left(\frac{143}{1278}\right)\) | \(e\left(\frac{281}{639}\right)\) | \(e\left(\frac{2}{213}\right)\) | \(e\left(\frac{139}{639}\right)\) | \(e\left(\frac{625}{639}\right)\) | \(e\left(\frac{85}{426}\right)\) |
| \(\chi_{1279}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{319}{639}\right)\) | \(e\left(\frac{125}{1278}\right)\) | \(e\left(\frac{638}{639}\right)\) | \(e\left(\frac{4}{639}\right)\) | \(e\left(\frac{763}{1278}\right)\) | \(e\left(\frac{409}{639}\right)\) | \(e\left(\frac{106}{213}\right)\) | \(e\left(\frac{125}{639}\right)\) | \(e\left(\frac{323}{639}\right)\) | \(e\left(\frac{245}{426}\right)\) |
| \(\chi_{1279}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{430}{639}\right)\) | \(e\left(\frac{491}{1278}\right)\) | \(e\left(\frac{221}{639}\right)\) | \(e\left(\frac{394}{639}\right)\) | \(e\left(\frac{73}{1278}\right)\) | \(e\left(\frac{349}{639}\right)\) | \(e\left(\frac{4}{213}\right)\) | \(e\left(\frac{491}{639}\right)\) | \(e\left(\frac{185}{639}\right)\) | \(e\left(\frac{383}{426}\right)\) |
| \(\chi_{1279}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{639}\right)\) | \(e\left(\frac{949}{1278}\right)\) | \(e\left(\frac{202}{639}\right)\) | \(e\left(\frac{470}{639}\right)\) | \(e\left(\frac{1151}{1278}\right)\) | \(e\left(\frac{452}{639}\right)\) | \(e\left(\frac{101}{213}\right)\) | \(e\left(\frac{310}{639}\right)\) | \(e\left(\frac{571}{639}\right)\) | \(e\left(\frac{139}{426}\right)\) |
| \(\chi_{1279}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{639}\right)\) | \(e\left(\frac{1123}{1278}\right)\) | \(e\left(\frac{538}{639}\right)\) | \(e\left(\frac{404}{639}\right)\) | \(e\left(\frac{383}{1278}\right)\) | \(e\left(\frac{413}{639}\right)\) | \(e\left(\frac{56}{213}\right)\) | \(e\left(\frac{484}{639}\right)\) | \(e\left(\frac{34}{639}\right)\) | \(e\left(\frac{37}{426}\right)\) |
| \(\chi_{1279}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{232}{639}\right)\) | \(e\left(\frac{149}{1278}\right)\) | \(e\left(\frac{464}{639}\right)\) | \(e\left(\frac{61}{639}\right)\) | \(e\left(\frac{613}{1278}\right)\) | \(e\left(\frac{7}{639}\right)\) | \(e\left(\frac{19}{213}\right)\) | \(e\left(\frac{149}{639}\right)\) | \(e\left(\frac{293}{639}\right)\) | \(e\left(\frac{275}{426}\right)\) |
| \(\chi_{1279}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{639}\right)\) | \(e\left(\frac{185}{1278}\right)\) | \(e\left(\frac{203}{639}\right)\) | \(e\left(\frac{466}{639}\right)\) | \(e\left(\frac{1027}{1278}\right)\) | \(e\left(\frac{43}{639}\right)\) | \(e\left(\frac{208}{213}\right)\) | \(e\left(\frac{185}{639}\right)\) | \(e\left(\frac{248}{639}\right)\) | \(e\left(\frac{107}{426}\right)\) |
| \(\chi_{1279}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{488}{639}\right)\) | \(e\left(\frac{49}{1278}\right)\) | \(e\left(\frac{337}{639}\right)\) | \(e\left(\frac{569}{639}\right)\) | \(e\left(\frac{1025}{1278}\right)\) | \(e\left(\frac{191}{639}\right)\) | \(e\left(\frac{62}{213}\right)\) | \(e\left(\frac{49}{639}\right)\) | \(e\left(\frac{418}{639}\right)\) | \(e\left(\frac{79}{426}\right)\) |
| \(\chi_{1279}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{515}{639}\right)\) | \(e\left(\frac{967}{1278}\right)\) | \(e\left(\frac{391}{639}\right)\) | \(e\left(\frac{353}{639}\right)\) | \(e\left(\frac{719}{1278}\right)\) | \(e\left(\frac{470}{639}\right)\) | \(e\left(\frac{89}{213}\right)\) | \(e\left(\frac{328}{639}\right)\) | \(e\left(\frac{229}{639}\right)\) | \(e\left(\frac{55}{426}\right)\) |
| \(\chi_{1279}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{639}\right)\) | \(e\left(\frac{1003}{1278}\right)\) | \(e\left(\frac{130}{639}\right)\) | \(e\left(\frac{119}{639}\right)\) | \(e\left(\frac{1133}{1278}\right)\) | \(e\left(\frac{506}{639}\right)\) | \(e\left(\frac{65}{213}\right)\) | \(e\left(\frac{364}{639}\right)\) | \(e\left(\frac{184}{639}\right)\) | \(e\left(\frac{313}{426}\right)\) |
| \(\chi_{1279}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{634}{639}\right)\) | \(e\left(\frac{611}{1278}\right)\) | \(e\left(\frac{629}{639}\right)\) | \(e\left(\frac{40}{639}\right)\) | \(e\left(\frac{601}{1278}\right)\) | \(e\left(\frac{256}{639}\right)\) | \(e\left(\frac{208}{213}\right)\) | \(e\left(\frac{611}{639}\right)\) | \(e\left(\frac{35}{639}\right)\) | \(e\left(\frac{107}{426}\right)\) |
| \(\chi_{1279}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{520}{639}\right)\) | \(e\left(\frac{995}{1278}\right)\) | \(e\left(\frac{401}{639}\right)\) | \(e\left(\frac{313}{639}\right)\) | \(e\left(\frac{757}{1278}\right)\) | \(e\left(\frac{214}{639}\right)\) | \(e\left(\frac{94}{213}\right)\) | \(e\left(\frac{356}{639}\right)\) | \(e\left(\frac{194}{639}\right)\) | \(e\left(\frac{161}{426}\right)\) |
| \(\chi_{1279}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{382}{639}\right)\) | \(e\left(\frac{989}{1278}\right)\) | \(e\left(\frac{125}{639}\right)\) | \(e\left(\frac{139}{639}\right)\) | \(e\left(\frac{475}{1278}\right)\) | \(e\left(\frac{634}{639}\right)\) | \(e\left(\frac{169}{213}\right)\) | \(e\left(\frac{350}{639}\right)\) | \(e\left(\frac{521}{639}\right)\) | \(e\left(\frac{47}{426}\right)\) |
| \(\chi_{1279}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{592}{639}\right)\) | \(e\left(\frac{887}{1278}\right)\) | \(e\left(\frac{545}{639}\right)\) | \(e\left(\frac{376}{639}\right)\) | \(e\left(\frac{793}{1278}\right)\) | \(e\left(\frac{106}{639}\right)\) | \(e\left(\frac{166}{213}\right)\) | \(e\left(\frac{248}{639}\right)\) | \(e\left(\frac{329}{639}\right)\) | \(e\left(\frac{239}{426}\right)\) |
| \(\chi_{1279}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{328}{639}\right)\) | \(e\left(\frac{431}{1278}\right)\) | \(e\left(\frac{17}{639}\right)\) | \(e\left(\frac{571}{639}\right)\) | \(e\left(\frac{1087}{1278}\right)\) | \(e\left(\frac{76}{639}\right)\) | \(e\left(\frac{115}{213}\right)\) | \(e\left(\frac{431}{639}\right)\) | \(e\left(\frac{260}{639}\right)\) | \(e\left(\frac{95}{426}\right)\) |
| \(\chi_{1279}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{639}\right)\) | \(e\left(\frac{1169}{1278}\right)\) | \(e\left(\frac{98}{639}\right)\) | \(e\left(\frac{247}{639}\right)\) | \(e\left(\frac{1267}{1278}\right)\) | \(e\left(\frac{175}{639}\right)\) | \(e\left(\frac{49}{213}\right)\) | \(e\left(\frac{530}{639}\right)\) | \(e\left(\frac{296}{639}\right)\) | \(e\left(\frac{59}{426}\right)\) |
| \(\chi_{1279}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{434}{639}\right)\) | \(e\left(\frac{769}{1278}\right)\) | \(e\left(\frac{229}{639}\right)\) | \(e\left(\frac{362}{639}\right)\) | \(e\left(\frac{359}{1278}\right)\) | \(e\left(\frac{272}{639}\right)\) | \(e\left(\frac{8}{213}\right)\) | \(e\left(\frac{130}{639}\right)\) | \(e\left(\frac{157}{639}\right)\) | \(e\left(\frac{127}{426}\right)\) |
| \(\chi_{1279}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{623}{639}\right)\) | \(e\left(\frac{805}{1278}\right)\) | \(e\left(\frac{607}{639}\right)\) | \(e\left(\frac{128}{639}\right)\) | \(e\left(\frac{773}{1278}\right)\) | \(e\left(\frac{308}{639}\right)\) | \(e\left(\frac{197}{213}\right)\) | \(e\left(\frac{166}{639}\right)\) | \(e\left(\frac{112}{639}\right)\) | \(e\left(\frac{385}{426}\right)\) |
| \(\chi_{1279}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{514}{639}\right)\) | \(e\left(\frac{1217}{1278}\right)\) | \(e\left(\frac{389}{639}\right)\) | \(e\left(\frac{361}{639}\right)\) | \(e\left(\frac{967}{1278}\right)\) | \(e\left(\frac{10}{639}\right)\) | \(e\left(\frac{88}{213}\right)\) | \(e\left(\frac{578}{639}\right)\) | \(e\left(\frac{236}{639}\right)\) | \(e\left(\frac{119}{426}\right)\) |
| \(\chi_{1279}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{639}\right)\) | \(e\left(\frac{791}{1278}\right)\) | \(e\left(\frac{602}{639}\right)\) | \(e\left(\frac{148}{639}\right)\) | \(e\left(\frac{115}{1278}\right)\) | \(e\left(\frac{436}{639}\right)\) | \(e\left(\frac{88}{213}\right)\) | \(e\left(\frac{152}{639}\right)\) | \(e\left(\frac{449}{639}\right)\) | \(e\left(\frac{119}{426}\right)\) |
| \(\chi_{1279}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{332}{639}\right)\) | \(e\left(\frac{709}{1278}\right)\) | \(e\left(\frac{25}{639}\right)\) | \(e\left(\frac{539}{639}\right)\) | \(e\left(\frac{95}{1278}\right)\) | \(e\left(\frac{638}{639}\right)\) | \(e\left(\frac{119}{213}\right)\) | \(e\left(\frac{70}{639}\right)\) | \(e\left(\frac{232}{639}\right)\) | \(e\left(\frac{265}{426}\right)\) |
| \(\chi_{1279}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{639}\right)\) | \(e\left(\frac{59}{1278}\right)\) | \(e\left(\frac{158}{639}\right)\) | \(e\left(\frac{7}{639}\right)\) | \(e\left(\frac{217}{1278}\right)\) | \(e\left(\frac{556}{639}\right)\) | \(e\left(\frac{79}{213}\right)\) | \(e\left(\frac{59}{639}\right)\) | \(e\left(\frac{86}{639}\right)\) | \(e\left(\frac{269}{426}\right)\) |
| \(\chi_{1279}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{370}{639}\right)\) | \(e\left(\frac{155}{1278}\right)\) | \(e\left(\frac{101}{639}\right)\) | \(e\left(\frac{235}{639}\right)\) | \(e\left(\frac{895}{1278}\right)\) | \(e\left(\frac{226}{639}\right)\) | \(e\left(\frac{157}{213}\right)\) | \(e\left(\frac{155}{639}\right)\) | \(e\left(\frac{605}{639}\right)\) | \(e\left(\frac{389}{426}\right)\) |
| \(\chi_{1279}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{639}\right)\) | \(e\left(\frac{235}{1278}\right)\) | \(e\left(\frac{586}{639}\right)\) | \(e\left(\frac{212}{639}\right)\) | \(e\left(\frac{821}{1278}\right)\) | \(e\left(\frac{590}{639}\right)\) | \(e\left(\frac{80}{213}\right)\) | \(e\left(\frac{235}{639}\right)\) | \(e\left(\frac{505}{639}\right)\) | \(e\left(\frac{205}{426}\right)\) |
| \(\chi_{1279}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{518}{639}\right)\) | \(e\left(\frac{217}{1278}\right)\) | \(e\left(\frac{397}{639}\right)\) | \(e\left(\frac{329}{639}\right)\) | \(e\left(\frac{1253}{1278}\right)\) | \(e\left(\frac{572}{639}\right)\) | \(e\left(\frac{92}{213}\right)\) | \(e\left(\frac{217}{639}\right)\) | \(e\left(\frac{208}{639}\right)\) | \(e\left(\frac{289}{426}\right)\) |