from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1039, base_ring=CyclotomicField(1038))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1039))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1039\) | |
| Conductor: | \(1039\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1038\) | 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: | $\Q(\zeta_{519})$ |
| Fixed field: | Number field defined by a degree 1038 polynomial (not computed) |
First 31 of 344 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1039}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{116}{519}\right)\) | \(e\left(\frac{1}{1038}\right)\) | \(e\left(\frac{232}{519}\right)\) | \(e\left(\frac{127}{173}\right)\) | \(e\left(\frac{233}{1038}\right)\) | \(e\left(\frac{79}{519}\right)\) | \(e\left(\frac{116}{173}\right)\) | \(e\left(\frac{1}{519}\right)\) | \(e\left(\frac{497}{519}\right)\) | \(e\left(\frac{877}{1038}\right)\) |
| \(\chi_{1039}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{519}\right)\) | \(e\left(\frac{233}{1038}\right)\) | \(e\left(\frac{80}{519}\right)\) | \(e\left(\frac{8}{173}\right)\) | \(e\left(\frac{313}{1038}\right)\) | \(e\left(\frac{242}{519}\right)\) | \(e\left(\frac{40}{173}\right)\) | \(e\left(\frac{233}{519}\right)\) | \(e\left(\frac{64}{519}\right)\) | \(e\left(\frac{893}{1038}\right)\) |
| \(\chi_{1039}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{519}\right)\) | \(e\left(\frac{877}{1038}\right)\) | \(e\left(\frac{16}{519}\right)\) | \(e\left(\frac{140}{173}\right)\) | \(e\left(\frac{893}{1038}\right)\) | \(e\left(\frac{256}{519}\right)\) | \(e\left(\frac{8}{173}\right)\) | \(e\left(\frac{358}{519}\right)\) | \(e\left(\frac{428}{519}\right)\) | \(e\left(\frac{1009}{1038}\right)\) |
| \(\chi_{1039}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{278}{519}\right)\) | \(e\left(\frac{763}{1038}\right)\) | \(e\left(\frac{37}{519}\right)\) | \(e\left(\frac{21}{173}\right)\) | \(e\left(\frac{281}{1038}\right)\) | \(e\left(\frac{73}{519}\right)\) | \(e\left(\frac{105}{173}\right)\) | \(e\left(\frac{244}{519}\right)\) | \(e\left(\frac{341}{519}\right)\) | \(e\left(\frac{679}{1038}\right)\) |
| \(\chi_{1039}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{451}{519}\right)\) | \(e\left(\frac{71}{1038}\right)\) | \(e\left(\frac{383}{519}\right)\) | \(e\left(\frac{21}{173}\right)\) | \(e\left(\frac{973}{1038}\right)\) | \(e\left(\frac{419}{519}\right)\) | \(e\left(\frac{105}{173}\right)\) | \(e\left(\frac{71}{519}\right)\) | \(e\left(\frac{514}{519}\right)\) | \(e\left(\frac{1025}{1038}\right)\) |
| \(\chi_{1039}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{407}{519}\right)\) | \(e\left(\frac{697}{1038}\right)\) | \(e\left(\frac{295}{519}\right)\) | \(e\left(\frac{116}{173}\right)\) | \(e\left(\frac{473}{1038}\right)\) | \(e\left(\frac{49}{519}\right)\) | \(e\left(\frac{61}{173}\right)\) | \(e\left(\frac{178}{519}\right)\) | \(e\left(\frac{236}{519}\right)\) | \(e\left(\frac{925}{1038}\right)\) |
| \(\chi_{1039}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{202}{519}\right)\) | \(e\left(\frac{995}{1038}\right)\) | \(e\left(\frac{404}{519}\right)\) | \(e\left(\frac{75}{173}\right)\) | \(e\left(\frac{361}{1038}\right)\) | \(e\left(\frac{236}{519}\right)\) | \(e\left(\frac{29}{173}\right)\) | \(e\left(\frac{476}{519}\right)\) | \(e\left(\frac{427}{519}\right)\) | \(e\left(\frac{695}{1038}\right)\) |
| \(\chi_{1039}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{519}\right)\) | \(e\left(\frac{569}{1038}\right)\) | \(e\left(\frac{182}{519}\right)\) | \(e\left(\frac{122}{173}\right)\) | \(e\left(\frac{751}{1038}\right)\) | \(e\left(\frac{317}{519}\right)\) | \(e\left(\frac{91}{173}\right)\) | \(e\left(\frac{50}{519}\right)\) | \(e\left(\frac{457}{519}\right)\) | \(e\left(\frac{773}{1038}\right)\) |
| \(\chi_{1039}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{519}\right)\) | \(e\left(\frac{391}{1038}\right)\) | \(e\left(\frac{406}{519}\right)\) | \(e\left(\frac{6}{173}\right)\) | \(e\left(\frac{797}{1038}\right)\) | \(e\left(\frac{268}{519}\right)\) | \(e\left(\frac{30}{173}\right)\) | \(e\left(\frac{391}{519}\right)\) | \(e\left(\frac{221}{519}\right)\) | \(e\left(\frac{367}{1038}\right)\) |
| \(\chi_{1039}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{416}{519}\right)\) | \(e\left(\frac{451}{1038}\right)\) | \(e\left(\frac{313}{519}\right)\) | \(e\left(\frac{14}{173}\right)\) | \(e\left(\frac{245}{1038}\right)\) | \(e\left(\frac{337}{519}\right)\) | \(e\left(\frac{70}{173}\right)\) | \(e\left(\frac{451}{519}\right)\) | \(e\left(\frac{458}{519}\right)\) | \(e\left(\frac{49}{1038}\right)\) |
| \(\chi_{1039}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{519}\right)\) | \(e\left(\frac{929}{1038}\right)\) | \(e\left(\frac{143}{519}\right)\) | \(e\left(\frac{170}{173}\right)\) | \(e\left(\frac{553}{1038}\right)\) | \(e\left(\frac{212}{519}\right)\) | \(e\left(\frac{158}{173}\right)\) | \(e\left(\frac{410}{519}\right)\) | \(e\left(\frac{322}{519}\right)\) | \(e\left(\frac{941}{1038}\right)\) |
| \(\chi_{1039}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{519}\right)\) | \(e\left(\frac{1001}{1038}\right)\) | \(e\left(\frac{239}{519}\right)\) | \(e\left(\frac{145}{173}\right)\) | \(e\left(\frac{721}{1038}\right)\) | \(e\left(\frac{191}{519}\right)\) | \(e\left(\frac{33}{173}\right)\) | \(e\left(\frac{482}{519}\right)\) | \(e\left(\frac{295}{519}\right)\) | \(e\left(\frac{767}{1038}\right)\) |
| \(\chi_{1039}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{519}\right)\) | \(e\left(\frac{1009}{1038}\right)\) | \(e\left(\frac{19}{519}\right)\) | \(e\left(\frac{123}{173}\right)\) | \(e\left(\frac{509}{1038}\right)\) | \(e\left(\frac{304}{519}\right)\) | \(e\left(\frac{96}{173}\right)\) | \(e\left(\frac{490}{519}\right)\) | \(e\left(\frac{119}{519}\right)\) | \(e\left(\frac{517}{1038}\right)\) |
| \(\chi_{1039}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{272}{519}\right)\) | \(e\left(\frac{235}{1038}\right)\) | \(e\left(\frac{25}{519}\right)\) | \(e\left(\frac{89}{173}\right)\) | \(e\left(\frac{779}{1038}\right)\) | \(e\left(\frac{400}{519}\right)\) | \(e\left(\frac{99}{173}\right)\) | \(e\left(\frac{235}{519}\right)\) | \(e\left(\frac{20}{519}\right)\) | \(e\left(\frac{571}{1038}\right)\) |
| \(\chi_{1039}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{170}{519}\right)\) | \(e\left(\frac{601}{1038}\right)\) | \(e\left(\frac{340}{519}\right)\) | \(e\left(\frac{34}{173}\right)\) | \(e\left(\frac{941}{1038}\right)\) | \(e\left(\frac{250}{519}\right)\) | \(e\left(\frac{170}{173}\right)\) | \(e\left(\frac{82}{519}\right)\) | \(e\left(\frac{272}{519}\right)\) | \(e\left(\frac{811}{1038}\right)\) |
| \(\chi_{1039}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{505}{519}\right)\) | \(e\left(\frac{671}{1038}\right)\) | \(e\left(\frac{491}{519}\right)\) | \(e\left(\frac{101}{173}\right)\) | \(e\left(\frac{643}{1038}\right)\) | \(e\left(\frac{71}{519}\right)\) | \(e\left(\frac{159}{173}\right)\) | \(e\left(\frac{152}{519}\right)\) | \(e\left(\frac{289}{519}\right)\) | \(e\left(\frac{959}{1038}\right)\) |
| \(\chi_{1039}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{519}\right)\) | \(e\left(\frac{893}{1038}\right)\) | \(e\left(\frac{95}{519}\right)\) | \(e\left(\frac{96}{173}\right)\) | \(e\left(\frac{469}{1038}\right)\) | \(e\left(\frac{482}{519}\right)\) | \(e\left(\frac{134}{173}\right)\) | \(e\left(\frac{374}{519}\right)\) | \(e\left(\frac{76}{519}\right)\) | \(e\left(\frac{509}{1038}\right)\) |
| \(\chi_{1039}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{176}{519}\right)\) | \(e\left(\frac{91}{1038}\right)\) | \(e\left(\frac{352}{519}\right)\) | \(e\left(\frac{139}{173}\right)\) | \(e\left(\frac{443}{1038}\right)\) | \(e\left(\frac{442}{519}\right)\) | \(e\left(\frac{3}{173}\right)\) | \(e\left(\frac{91}{519}\right)\) | \(e\left(\frac{74}{519}\right)\) | \(e\left(\frac{919}{1038}\right)\) |
| \(\chi_{1039}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{440}{519}\right)\) | \(e\left(\frac{487}{1038}\right)\) | \(e\left(\frac{361}{519}\right)\) | \(e\left(\frac{88}{173}\right)\) | \(e\left(\frac{329}{1038}\right)\) | \(e\left(\frac{67}{519}\right)\) | \(e\left(\frac{94}{173}\right)\) | \(e\left(\frac{487}{519}\right)\) | \(e\left(\frac{185}{519}\right)\) | \(e\left(\frac{481}{1038}\right)\) |
| \(\chi_{1039}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{182}{519}\right)\) | \(e\left(\frac{619}{1038}\right)\) | \(e\left(\frac{364}{519}\right)\) | \(e\left(\frac{71}{173}\right)\) | \(e\left(\frac{983}{1038}\right)\) | \(e\left(\frac{115}{519}\right)\) | \(e\left(\frac{9}{173}\right)\) | \(e\left(\frac{100}{519}\right)\) | \(e\left(\frac{395}{519}\right)\) | \(e\left(\frac{1027}{1038}\right)\) |
| \(\chi_{1039}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{206}{519}\right)\) | \(e\left(\frac{655}{1038}\right)\) | \(e\left(\frac{412}{519}\right)\) | \(e\left(\frac{145}{173}\right)\) | \(e\left(\frac{29}{1038}\right)\) | \(e\left(\frac{364}{519}\right)\) | \(e\left(\frac{33}{173}\right)\) | \(e\left(\frac{136}{519}\right)\) | \(e\left(\frac{122}{519}\right)\) | \(e\left(\frac{421}{1038}\right)\) |
| \(\chi_{1039}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{519}\right)\) | \(e\left(\frac{623}{1038}\right)\) | \(e\left(\frac{254}{519}\right)\) | \(e\left(\frac{60}{173}\right)\) | \(e\left(\frac{877}{1038}\right)\) | \(e\left(\frac{431}{519}\right)\) | \(e\left(\frac{127}{173}\right)\) | \(e\left(\frac{104}{519}\right)\) | \(e\left(\frac{307}{519}\right)\) | \(e\left(\frac{383}{1038}\right)\) |
| \(\chi_{1039}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{519}\right)\) | \(e\left(\frac{535}{1038}\right)\) | \(e\left(\frac{79}{519}\right)\) | \(e\left(\frac{129}{173}\right)\) | \(e\left(\frac{95}{1038}\right)\) | \(e\left(\frac{226}{519}\right)\) | \(e\left(\frac{126}{173}\right)\) | \(e\left(\frac{16}{519}\right)\) | \(e\left(\frac{167}{519}\right)\) | \(e\left(\frac{19}{1038}\right)\) |
| \(\chi_{1039}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{519}\right)\) | \(e\left(\frac{869}{1038}\right)\) | \(e\left(\frac{236}{519}\right)\) | \(e\left(\frac{162}{173}\right)\) | \(e\left(\frac{67}{1038}\right)\) | \(e\left(\frac{143}{519}\right)\) | \(e\left(\frac{118}{173}\right)\) | \(e\left(\frac{350}{519}\right)\) | \(e\left(\frac{85}{519}\right)\) | \(e\left(\frac{221}{1038}\right)\) |
| \(\chi_{1039}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{340}{519}\right)\) | \(e\left(\frac{683}{1038}\right)\) | \(e\left(\frac{161}{519}\right)\) | \(e\left(\frac{68}{173}\right)\) | \(e\left(\frac{325}{1038}\right)\) | \(e\left(\frac{500}{519}\right)\) | \(e\left(\frac{167}{173}\right)\) | \(e\left(\frac{164}{519}\right)\) | \(e\left(\frac{25}{519}\right)\) | \(e\left(\frac{65}{1038}\right)\) |
| \(\chi_{1039}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{515}{519}\right)\) | \(e\left(\frac{859}{1038}\right)\) | \(e\left(\frac{511}{519}\right)\) | \(e\left(\frac{103}{173}\right)\) | \(e\left(\frac{851}{1038}\right)\) | \(e\left(\frac{391}{519}\right)\) | \(e\left(\frac{169}{173}\right)\) | \(e\left(\frac{340}{519}\right)\) | \(e\left(\frac{305}{519}\right)\) | \(e\left(\frac{793}{1038}\right)\) |
| \(\chi_{1039}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{146}{519}\right)\) | \(e\left(\frac{565}{1038}\right)\) | \(e\left(\frac{292}{519}\right)\) | \(e\left(\frac{133}{173}\right)\) | \(e\left(\frac{857}{1038}\right)\) | \(e\left(\frac{1}{519}\right)\) | \(e\left(\frac{146}{173}\right)\) | \(e\left(\frac{46}{519}\right)\) | \(e\left(\frac{26}{519}\right)\) | \(e\left(\frac{379}{1038}\right)\) |
| \(\chi_{1039}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{519}\right)\) | \(e\left(\frac{203}{1038}\right)\) | \(e\left(\frac{386}{519}\right)\) | \(e\left(\frac{4}{173}\right)\) | \(e\left(\frac{589}{1038}\right)\) | \(e\left(\frac{467}{519}\right)\) | \(e\left(\frac{20}{173}\right)\) | \(e\left(\frac{203}{519}\right)\) | \(e\left(\frac{205}{519}\right)\) | \(e\left(\frac{533}{1038}\right)\) |
| \(\chi_{1039}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{519}\right)\) | \(e\left(\frac{613}{1038}\right)\) | \(e\left(\frac{10}{519}\right)\) | \(e\left(\frac{1}{173}\right)\) | \(e\left(\frac{623}{1038}\right)\) | \(e\left(\frac{160}{519}\right)\) | \(e\left(\frac{5}{173}\right)\) | \(e\left(\frac{94}{519}\right)\) | \(e\left(\frac{8}{519}\right)\) | \(e\left(\frac{955}{1038}\right)\) |
| \(\chi_{1039}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{196}{519}\right)\) | \(e\left(\frac{467}{1038}\right)\) | \(e\left(\frac{392}{519}\right)\) | \(e\left(\frac{143}{173}\right)\) | \(e\left(\frac{859}{1038}\right)\) | \(e\left(\frac{44}{519}\right)\) | \(e\left(\frac{23}{173}\right)\) | \(e\left(\frac{467}{519}\right)\) | \(e\left(\frac{106}{519}\right)\) | \(e\left(\frac{587}{1038}\right)\) |
| \(\chi_{1039}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{519}\right)\) | \(e\left(\frac{7}{1038}\right)\) | \(e\left(\frac{67}{519}\right)\) | \(e\left(\frac{24}{173}\right)\) | \(e\left(\frac{593}{1038}\right)\) | \(e\left(\frac{34}{519}\right)\) | \(e\left(\frac{120}{173}\right)\) | \(e\left(\frac{7}{519}\right)\) | \(e\left(\frac{365}{519}\right)\) | \(e\left(\frac{949}{1038}\right)\) |