from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2303, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([23,420]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,2303))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2303\) | |
| Conductor: | \(2303\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(966\) | 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: | $\Q(\zeta_{483})$ |
| Fixed field: | Number field defined by a degree 966 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2303}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{483}\right)\) | \(e\left(\frac{695}{966}\right)\) | \(e\left(\frac{430}{483}\right)\) | \(e\left(\frac{121}{966}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{212}{483}\right)\) | \(e\left(\frac{551}{966}\right)\) | \(e\left(\frac{481}{483}\right)\) | \(e\left(\frac{589}{966}\right)\) |
| \(\chi_{2303}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{483}\right)\) | \(e\left(\frac{589}{966}\right)\) | \(e\left(\frac{215}{483}\right)\) | \(e\left(\frac{785}{966}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{106}{483}\right)\) | \(e\left(\frac{517}{966}\right)\) | \(e\left(\frac{482}{483}\right)\) | \(e\left(\frac{53}{966}\right)\) |
| \(\chi_{2303}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{356}{483}\right)\) | \(e\left(\frac{533}{966}\right)\) | \(e\left(\frac{229}{483}\right)\) | \(e\left(\frac{589}{966}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{50}{483}\right)\) | \(e\left(\frac{335}{966}\right)\) | \(e\left(\frac{118}{483}\right)\) | \(e\left(\frac{25}{966}\right)\) |
| \(\chi_{2303}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{416}{483}\right)\) | \(e\left(\frac{53}{966}\right)\) | \(e\left(\frac{349}{483}\right)\) | \(e\left(\frac{151}{966}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{53}{483}\right)\) | \(e\left(\frac{17}{966}\right)\) | \(e\left(\frac{241}{483}\right)\) | \(e\left(\frac{751}{966}\right)\) |
| \(\chi_{2303}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{483}\right)\) | \(e\left(\frac{583}{966}\right)\) | \(e\left(\frac{458}{483}\right)\) | \(e\left(\frac{695}{966}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{100}{483}\right)\) | \(e\left(\frac{187}{966}\right)\) | \(e\left(\frac{236}{483}\right)\) | \(e\left(\frac{533}{966}\right)\) |
| \(\chi_{2303}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{464}{483}\right)\) | \(e\left(\frac{635}{966}\right)\) | \(e\left(\frac{445}{483}\right)\) | \(e\left(\frac{187}{966}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{152}{483}\right)\) | \(e\left(\frac{149}{966}\right)\) | \(e\left(\frac{436}{483}\right)\) | \(e\left(\frac{559}{966}\right)\) |
| \(\chi_{2303}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{483}\right)\) | \(e\left(\frac{1}{966}\right)\) | \(e\left(\frac{362}{483}\right)\) | \(e\left(\frac{659}{966}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{1}{483}\right)\) | \(e\left(\frac{55}{966}\right)\) | \(e\left(\frac{41}{483}\right)\) | \(e\left(\frac{725}{966}\right)\) |
| \(\chi_{2303}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{483}\right)\) | \(e\left(\frac{937}{966}\right)\) | \(e\left(\frac{128}{483}\right)\) | \(e\left(\frac{209}{966}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{454}{483}\right)\) | \(e\left(\frac{337}{966}\right)\) | \(e\left(\frac{260}{483}\right)\) | \(e\left(\frac{227}{966}\right)\) |
| \(\chi_{2303}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{304}{483}\right)\) | \(e\left(\frac{949}{966}\right)\) | \(e\left(\frac{125}{483}\right)\) | \(e\left(\frac{389}{966}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{466}{483}\right)\) | \(e\left(\frac{31}{966}\right)\) | \(e\left(\frac{269}{483}\right)\) | \(e\left(\frac{233}{966}\right)\) |
| \(\chi_{2303}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{483}\right)\) | \(e\left(\frac{913}{966}\right)\) | \(e\left(\frac{134}{483}\right)\) | \(e\left(\frac{815}{966}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{67}{161}\right)\) | \(e\left(\frac{430}{483}\right)\) | \(e\left(\frac{949}{966}\right)\) | \(e\left(\frac{242}{483}\right)\) | \(e\left(\frac{215}{966}\right)\) |
| \(\chi_{2303}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{68}{483}\right)\) | \(e\left(\frac{905}{966}\right)\) | \(e\left(\frac{136}{483}\right)\) | \(e\left(\frac{373}{966}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{422}{483}\right)\) | \(e\left(\frac{509}{966}\right)\) | \(e\left(\frac{397}{483}\right)\) | \(e\left(\frac{211}{966}\right)\) |
| \(\chi_{2303}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{292}{483}\right)\) | \(e\left(\frac{79}{966}\right)\) | \(e\left(\frac{101}{483}\right)\) | \(e\left(\frac{863}{966}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{79}{483}\right)\) | \(e\left(\frac{481}{966}\right)\) | \(e\left(\frac{341}{483}\right)\) | \(e\left(\frac{281}{966}\right)\) |
| \(\chi_{2303}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{296}{483}\right)\) | \(e\left(\frac{47}{966}\right)\) | \(e\left(\frac{109}{483}\right)\) | \(e\left(\frac{61}{966}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{47}{483}\right)\) | \(e\left(\frac{653}{966}\right)\) | \(e\left(\frac{478}{483}\right)\) | \(e\left(\frac{265}{966}\right)\) |
| \(\chi_{2303}(110,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{475}{483}\right)\) | \(e\left(\frac{547}{966}\right)\) | \(e\left(\frac{467}{483}\right)\) | \(e\left(\frac{155}{966}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{153}{161}\right)\) | \(e\left(\frac{64}{483}\right)\) | \(e\left(\frac{139}{966}\right)\) | \(e\left(\frac{209}{483}\right)\) | \(e\left(\frac{515}{966}\right)\) |
| \(\chi_{2303}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{398}{483}\right)\) | \(e\left(\frac{197}{966}\right)\) | \(e\left(\frac{313}{483}\right)\) | \(e\left(\frac{379}{966}\right)\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{197}{483}\right)\) | \(e\left(\frac{209}{966}\right)\) | \(e\left(\frac{349}{483}\right)\) | \(e\left(\frac{823}{966}\right)\) |
| \(\chi_{2303}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{248}{483}\right)\) | \(e\left(\frac{431}{966}\right)\) | \(e\left(\frac{13}{483}\right)\) | \(e\left(\frac{25}{966}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{431}{483}\right)\) | \(e\left(\frac{521}{966}\right)\) | \(e\left(\frac{283}{483}\right)\) | \(e\left(\frac{457}{966}\right)\) |
| \(\chi_{2303}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{394}{483}\right)\) | \(e\left(\frac{229}{966}\right)\) | \(e\left(\frac{305}{483}\right)\) | \(e\left(\frac{215}{966}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{229}{483}\right)\) | \(e\left(\frac{37}{966}\right)\) | \(e\left(\frac{212}{483}\right)\) | \(e\left(\frac{839}{966}\right)\) |
| \(\chi_{2303}(136,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{74}{483}\right)\) | \(e\left(\frac{857}{966}\right)\) | \(e\left(\frac{148}{483}\right)\) | \(e\left(\frac{619}{966}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{374}{483}\right)\) | \(e\left(\frac{767}{966}\right)\) | \(e\left(\frac{361}{483}\right)\) | \(e\left(\frac{187}{966}\right)\) |
| \(\chi_{2303}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{483}\right)\) | \(e\left(\frac{545}{966}\right)\) | \(e\left(\frac{226}{483}\right)\) | \(e\left(\frac{769}{966}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{62}{483}\right)\) | \(e\left(\frac{29}{966}\right)\) | \(e\left(\frac{127}{483}\right)\) | \(e\left(\frac{31}{966}\right)\) |
| \(\chi_{2303}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{88}{483}\right)\) | \(e\left(\frac{745}{966}\right)\) | \(e\left(\frac{176}{483}\right)\) | \(e\left(\frac{227}{966}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{262}{483}\right)\) | \(e\left(\frac{403}{966}\right)\) | \(e\left(\frac{116}{483}\right)\) | \(e\left(\frac{131}{966}\right)\) |
| \(\chi_{2303}(150,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{483}\right)\) | \(e\left(\frac{401}{966}\right)\) | \(e\left(\frac{262}{483}\right)\) | \(e\left(\frac{541}{966}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{401}{483}\right)\) | \(e\left(\frac{803}{966}\right)\) | \(e\left(\frac{19}{483}\right)\) | \(e\left(\frac{925}{966}\right)\) |
| \(\chi_{2303}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{483}\right)\) | \(e\left(\frac{593}{966}\right)\) | \(e\left(\frac{214}{483}\right)\) | \(e\left(\frac{523}{966}\right)\) | \(e\left(\frac{269}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{110}{483}\right)\) | \(e\left(\frac{737}{966}\right)\) | \(e\left(\frac{163}{483}\right)\) | \(e\left(\frac{55}{966}\right)\) |
| \(\chi_{2303}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{244}{483}\right)\) | \(e\left(\frac{463}{966}\right)\) | \(e\left(\frac{5}{483}\right)\) | \(e\left(\frac{827}{966}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{463}{483}\right)\) | \(e\left(\frac{349}{966}\right)\) | \(e\left(\frac{146}{483}\right)\) | \(e\left(\frac{473}{966}\right)\) |
| \(\chi_{2303}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{358}{483}\right)\) | \(e\left(\frac{517}{966}\right)\) | \(e\left(\frac{233}{483}\right)\) | \(e\left(\frac{671}{966}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{34}{483}\right)\) | \(e\left(\frac{421}{966}\right)\) | \(e\left(\frac{428}{483}\right)\) | \(e\left(\frac{17}{966}\right)\) |
| \(\chi_{2303}(192,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{483}\right)\) | \(e\left(\frac{377}{966}\right)\) | \(e\left(\frac{268}{483}\right)\) | \(e\left(\frac{181}{966}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{377}{483}\right)\) | \(e\left(\frac{449}{966}\right)\) | \(e\left(\frac{1}{483}\right)\) | \(e\left(\frac{913}{966}\right)\) |
| \(\chi_{2303}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{466}{483}\right)\) | \(e\left(\frac{619}{966}\right)\) | \(e\left(\frac{449}{483}\right)\) | \(e\left(\frac{269}{966}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{136}{483}\right)\) | \(e\left(\frac{235}{966}\right)\) | \(e\left(\frac{263}{483}\right)\) | \(e\left(\frac{551}{966}\right)\) |
| \(\chi_{2303}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{483}\right)\) | \(e\left(\frac{509}{966}\right)\) | \(e\left(\frac{235}{483}\right)\) | \(e\left(\frac{229}{966}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{26}{483}\right)\) | \(e\left(\frac{947}{966}\right)\) | \(e\left(\frac{100}{483}\right)\) | \(e\left(\frac{13}{966}\right)\) |
| \(\chi_{2303}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{483}\right)\) | \(e\left(\frac{449}{966}\right)\) | \(e\left(\frac{250}{483}\right)\) | \(e\left(\frac{295}{966}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{449}{483}\right)\) | \(e\left(\frac{545}{966}\right)\) | \(e\left(\frac{55}{483}\right)\) | \(e\left(\frac{949}{966}\right)\) |
| \(\chi_{2303}(220,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{483}\right)\) | \(e\left(\frac{11}{966}\right)\) | \(e\left(\frac{118}{483}\right)\) | \(e\left(\frac{487}{966}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{11}{483}\right)\) | \(e\left(\frac{605}{966}\right)\) | \(e\left(\frac{451}{483}\right)\) | \(e\left(\frac{247}{966}\right)\) |
| \(\chi_{2303}(222,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{400}{483}\right)\) | \(e\left(\frac{181}{966}\right)\) | \(e\left(\frac{317}{483}\right)\) | \(e\left(\frac{461}{966}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{78}{161}\right)\) | \(e\left(\frac{181}{483}\right)\) | \(e\left(\frac{295}{966}\right)\) | \(e\left(\frac{176}{483}\right)\) | \(e\left(\frac{815}{966}\right)\) |
| \(\chi_{2303}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{483}\right)\) | \(e\left(\frac{251}{966}\right)\) | \(e\left(\frac{58}{483}\right)\) | \(e\left(\frac{223}{966}\right)\) | \(e\left(\frac{103}{322}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{251}{483}\right)\) | \(e\left(\frac{281}{966}\right)\) | \(e\left(\frac{148}{483}\right)\) | \(e\left(\frac{367}{966}\right)\) |