from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4024, base_ring=CyclotomicField(502))
M = H._module
chi = DirichletCharacter(H, M([251,251,237]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,4024))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4024\) | |
| Conductor: | \(4024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(502\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{251})$ |
| Fixed field: | Number field defined by a degree 502 polynomial (not computed) |
First 31 of 250 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4024}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{251}\right)\) | \(e\left(\frac{244}{251}\right)\) | \(e\left(\frac{51}{502}\right)\) | \(e\left(\frac{75}{251}\right)\) | \(e\left(\frac{208}{251}\right)\) | \(e\left(\frac{21}{502}\right)\) | \(e\left(\frac{156}{251}\right)\) | \(e\left(\frac{57}{502}\right)\) | \(e\left(\frac{447}{502}\right)\) | \(e\left(\frac{377}{502}\right)\) |
| \(\chi_{4024}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{251}\right)\) | \(e\left(\frac{169}{251}\right)\) | \(e\left(\frac{203}{502}\right)\) | \(e\left(\frac{18}{251}\right)\) | \(e\left(\frac{70}{251}\right)\) | \(e\left(\frac{497}{502}\right)\) | \(e\left(\frac{178}{251}\right)\) | \(e\left(\frac{345}{502}\right)\) | \(e\left(\frac{37}{502}\right)\) | \(e\left(\frac{221}{502}\right)\) |
| \(\chi_{4024}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{251}\right)\) | \(e\left(\frac{67}{251}\right)\) | \(e\left(\frac{229}{502}\right)\) | \(e\left(\frac{71}{251}\right)\) | \(e\left(\frac{53}{251}\right)\) | \(e\left(\frac{301}{502}\right)\) | \(e\left(\frac{228}{251}\right)\) | \(e\left(\frac{315}{502}\right)\) | \(e\left(\frac{383}{502}\right)\) | \(e\left(\frac{49}{502}\right)\) |
| \(\chi_{4024}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{251}\right)\) | \(e\left(\frac{128}{251}\right)\) | \(e\left(\frac{179}{502}\right)\) | \(e\left(\frac{27}{251}\right)\) | \(e\left(\frac{105}{251}\right)\) | \(e\left(\frac{369}{502}\right)\) | \(e\left(\frac{16}{251}\right)\) | \(e\left(\frac{141}{502}\right)\) | \(e\left(\frac{181}{502}\right)\) | \(e\left(\frac{457}{502}\right)\) |
| \(\chi_{4024}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{242}{251}\right)\) | \(e\left(\frac{82}{251}\right)\) | \(e\left(\frac{299}{502}\right)\) | \(e\left(\frac{233}{251}\right)\) | \(e\left(\frac{181}{251}\right)\) | \(e\left(\frac{5}{502}\right)\) | \(e\left(\frac{73}{251}\right)\) | \(e\left(\frac{157}{502}\right)\) | \(e\left(\frac{465}{502}\right)\) | \(e\left(\frac{281}{502}\right)\) |
| \(\chi_{4024}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{251}\right)\) | \(e\left(\frac{70}{251}\right)\) | \(e\left(\frac{243}{502}\right)\) | \(e\left(\frac{3}{251}\right)\) | \(e\left(\frac{179}{251}\right)\) | \(e\left(\frac{41}{502}\right)\) | \(e\left(\frac{197}{251}\right)\) | \(e\left(\frac{183}{502}\right)\) | \(e\left(\frac{299}{502}\right)\) | \(e\left(\frac{497}{502}\right)\) |
| \(\chi_{4024}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{202}{251}\right)\) | \(e\left(\frac{56}{251}\right)\) | \(e\left(\frac{345}{502}\right)\) | \(e\left(\frac{153}{251}\right)\) | \(e\left(\frac{93}{251}\right)\) | \(e\left(\frac{83}{502}\right)\) | \(e\left(\frac{7}{251}\right)\) | \(e\left(\frac{297}{502}\right)\) | \(e\left(\frac{189}{502}\right)\) | \(e\left(\frac{247}{502}\right)\) |
| \(\chi_{4024}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{251}\right)\) | \(e\left(\frac{66}{251}\right)\) | \(e\left(\frac{57}{502}\right)\) | \(e\left(\frac{10}{251}\right)\) | \(e\left(\frac{11}{251}\right)\) | \(e\left(\frac{53}{502}\right)\) | \(e\left(\frac{71}{251}\right)\) | \(e\left(\frac{359}{502}\right)\) | \(e\left(\frac{411}{502}\right)\) | \(e\left(\frac{67}{502}\right)\) |
| \(\chi_{4024}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{251}\right)\) | \(e\left(\frac{149}{251}\right)\) | \(e\left(\frac{277}{502}\right)\) | \(e\left(\frac{53}{251}\right)\) | \(e\left(\frac{234}{251}\right)\) | \(e\left(\frac{55}{502}\right)\) | \(e\left(\frac{50}{251}\right)\) | \(e\left(\frac{221}{502}\right)\) | \(e\left(\frac{95}{502}\right)\) | \(e\left(\frac{79}{502}\right)\) |
| \(\chi_{4024}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{251}\right)\) | \(e\left(\frac{12}{251}\right)\) | \(e\left(\frac{307}{502}\right)\) | \(e\left(\frac{230}{251}\right)\) | \(e\left(\frac{2}{251}\right)\) | \(e\left(\frac{215}{502}\right)\) | \(e\left(\frac{127}{251}\right)\) | \(e\left(\frac{225}{502}\right)\) | \(e\left(\frac{417}{502}\right)\) | \(e\left(\frac{35}{502}\right)\) |
| \(\chi_{4024}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{251}\right)\) | \(e\left(\frac{10}{251}\right)\) | \(e\left(\frac{465}{502}\right)\) | \(e\left(\frac{108}{251}\right)\) | \(e\left(\frac{169}{251}\right)\) | \(e\left(\frac{221}{502}\right)\) | \(e\left(\frac{64}{251}\right)\) | \(e\left(\frac{313}{502}\right)\) | \(e\left(\frac{473}{502}\right)\) | \(e\left(\frac{71}{502}\right)\) |
| \(\chi_{4024}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{251}\right)\) | \(e\left(\frac{15}{251}\right)\) | \(e\left(\frac{321}{502}\right)\) | \(e\left(\frac{162}{251}\right)\) | \(e\left(\frac{128}{251}\right)\) | \(e\left(\frac{457}{502}\right)\) | \(e\left(\frac{96}{251}\right)\) | \(e\left(\frac{93}{502}\right)\) | \(e\left(\frac{333}{502}\right)\) | \(e\left(\frac{483}{502}\right)\) |
| \(\chi_{4024}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{162}{251}\right)\) | \(e\left(\frac{30}{251}\right)\) | \(e\left(\frac{391}{502}\right)\) | \(e\left(\frac{73}{251}\right)\) | \(e\left(\frac{5}{251}\right)\) | \(e\left(\frac{161}{502}\right)\) | \(e\left(\frac{192}{251}\right)\) | \(e\left(\frac{437}{502}\right)\) | \(e\left(\frac{415}{502}\right)\) | \(e\left(\frac{213}{502}\right)\) |
| \(\chi_{4024}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{251}\right)\) | \(e\left(\frac{211}{251}\right)\) | \(e\left(\frac{399}{502}\right)\) | \(e\left(\frac{70}{251}\right)\) | \(e\left(\frac{77}{251}\right)\) | \(e\left(\frac{371}{502}\right)\) | \(e\left(\frac{246}{251}\right)\) | \(e\left(\frac{3}{502}\right)\) | \(e\left(\frac{367}{502}\right)\) | \(e\left(\frac{469}{502}\right)\) |
| \(\chi_{4024}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{170}{251}\right)\) | \(e\left(\frac{236}{251}\right)\) | \(e\left(\frac{181}{502}\right)\) | \(e\left(\frac{89}{251}\right)\) | \(e\left(\frac{123}{251}\right)\) | \(e\left(\frac{45}{502}\right)\) | \(e\left(\frac{155}{251}\right)\) | \(e\left(\frac{409}{502}\right)\) | \(e\left(\frac{169}{502}\right)\) | \(e\left(\frac{19}{502}\right)\) |
| \(\chi_{4024}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{190}{251}\right)\) | \(e\left(\frac{249}{251}\right)\) | \(e\left(\frac{409}{502}\right)\) | \(e\left(\frac{129}{251}\right)\) | \(e\left(\frac{167}{251}\right)\) | \(e\left(\frac{257}{502}\right)\) | \(e\left(\frac{188}{251}\right)\) | \(e\left(\frac{339}{502}\right)\) | \(e\left(\frac{307}{502}\right)\) | \(e\left(\frac{287}{502}\right)\) |
| \(\chi_{4024}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{251}\right)\) | \(e\left(\frac{150}{251}\right)\) | \(e\left(\frac{449}{502}\right)\) | \(e\left(\frac{114}{251}\right)\) | \(e\left(\frac{25}{251}\right)\) | \(e\left(\frac{303}{502}\right)\) | \(e\left(\frac{207}{251}\right)\) | \(e\left(\frac{177}{502}\right)\) | \(e\left(\frac{67}{502}\right)\) | \(e\left(\frac{61}{502}\right)\) |
| \(\chi_{4024}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{251}\right)\) | \(e\left(\frac{103}{251}\right)\) | \(e\left(\frac{397}{502}\right)\) | \(e\left(\frac{8}{251}\right)\) | \(e\left(\frac{59}{251}\right)\) | \(e\left(\frac{193}{502}\right)\) | \(e\left(\frac{107}{251}\right)\) | \(e\left(\frac{237}{502}\right)\) | \(e\left(\frac{379}{502}\right)\) | \(e\left(\frac{405}{502}\right)\) |
| \(\chi_{4024}(267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{216}{251}\right)\) | \(e\left(\frac{40}{251}\right)\) | \(e\left(\frac{103}{502}\right)\) | \(e\left(\frac{181}{251}\right)\) | \(e\left(\frac{174}{251}\right)\) | \(e\left(\frac{131}{502}\right)\) | \(e\left(\frac{5}{251}\right)\) | \(e\left(\frac{499}{502}\right)\) | \(e\left(\frac{135}{502}\right)\) | \(e\left(\frac{33}{502}\right)\) |
| \(\chi_{4024}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{250}{251}\right)\) | \(e\left(\frac{37}{251}\right)\) | \(e\left(\frac{89}{502}\right)\) | \(e\left(\frac{249}{251}\right)\) | \(e\left(\frac{48}{251}\right)\) | \(e\left(\frac{391}{502}\right)\) | \(e\left(\frac{36}{251}\right)\) | \(e\left(\frac{129}{502}\right)\) | \(e\left(\frac{219}{502}\right)\) | \(e\left(\frac{87}{502}\right)\) |
| \(\chi_{4024}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{251}\right)\) | \(e\left(\frac{74}{251}\right)\) | \(e\left(\frac{429}{502}\right)\) | \(e\left(\frac{247}{251}\right)\) | \(e\left(\frac{96}{251}\right)\) | \(e\left(\frac{29}{502}\right)\) | \(e\left(\frac{72}{251}\right)\) | \(e\left(\frac{7}{502}\right)\) | \(e\left(\frac{187}{502}\right)\) | \(e\left(\frac{425}{502}\right)\) |
| \(\chi_{4024}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{251}\right)\) | \(e\left(\frac{235}{251}\right)\) | \(e\left(\frac{9}{502}\right)\) | \(e\left(\frac{28}{251}\right)\) | \(e\left(\frac{81}{251}\right)\) | \(e\left(\frac{299}{502}\right)\) | \(e\left(\frac{249}{251}\right)\) | \(e\left(\frac{453}{502}\right)\) | \(e\left(\frac{197}{502}\right)\) | \(e\left(\frac{37}{502}\right)\) |
| \(\chi_{4024}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{140}{251}\right)\) | \(e\left(\frac{91}{251}\right)\) | \(e\left(\frac{341}{502}\right)\) | \(e\left(\frac{29}{251}\right)\) | \(e\left(\frac{57}{251}\right)\) | \(e\left(\frac{229}{502}\right)\) | \(e\left(\frac{231}{251}\right)\) | \(e\left(\frac{263}{502}\right)\) | \(e\left(\frac{213}{502}\right)\) | \(e\left(\frac{119}{502}\right)\) |
| \(\chi_{4024}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{251}\right)\) | \(e\left(\frac{50}{251}\right)\) | \(e\left(\frac{317}{502}\right)\) | \(e\left(\frac{38}{251}\right)\) | \(e\left(\frac{92}{251}\right)\) | \(e\left(\frac{101}{502}\right)\) | \(e\left(\frac{69}{251}\right)\) | \(e\left(\frac{59}{502}\right)\) | \(e\left(\frac{357}{502}\right)\) | \(e\left(\frac{355}{502}\right)\) |
| \(\chi_{4024}(395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{246}{251}\right)\) | \(e\left(\frac{185}{251}\right)\) | \(e\left(\frac{445}{502}\right)\) | \(e\left(\frac{241}{251}\right)\) | \(e\left(\frac{240}{251}\right)\) | \(e\left(\frac{449}{502}\right)\) | \(e\left(\frac{180}{251}\right)\) | \(e\left(\frac{143}{502}\right)\) | \(e\left(\frac{91}{502}\right)\) | \(e\left(\frac{435}{502}\right)\) |
| \(\chi_{4024}(403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{251}\right)\) | \(e\left(\frac{203}{251}\right)\) | \(e\left(\frac{27}{502}\right)\) | \(e\left(\frac{84}{251}\right)\) | \(e\left(\frac{243}{251}\right)\) | \(e\left(\frac{395}{502}\right)\) | \(e\left(\frac{245}{251}\right)\) | \(e\left(\frac{355}{502}\right)\) | \(e\left(\frac{89}{502}\right)\) | \(e\left(\frac{111}{502}\right)\) |
| \(\chi_{4024}(411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{251}\right)\) | \(e\left(\frac{158}{251}\right)\) | \(e\left(\frac{319}{502}\right)\) | \(e\left(\frac{100}{251}\right)\) | \(e\left(\frac{110}{251}\right)\) | \(e\left(\frac{279}{502}\right)\) | \(e\left(\frac{208}{251}\right)\) | \(e\left(\frac{327}{502}\right)\) | \(e\left(\frac{345}{502}\right)\) | \(e\left(\frac{419}{502}\right)\) |
| \(\chi_{4024}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{188}{251}\right)\) | \(e\left(\frac{72}{251}\right)\) | \(e\left(\frac{85}{502}\right)\) | \(e\left(\frac{125}{251}\right)\) | \(e\left(\frac{12}{251}\right)\) | \(e\left(\frac{35}{502}\right)\) | \(e\left(\frac{9}{251}\right)\) | \(e\left(\frac{95}{502}\right)\) | \(e\left(\frac{243}{502}\right)\) | \(e\left(\frac{461}{502}\right)\) |
| \(\chi_{4024}(451,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{251}\right)\) | \(e\left(\frac{13}{251}\right)\) | \(e\left(\frac{479}{502}\right)\) | \(e\left(\frac{40}{251}\right)\) | \(e\left(\frac{44}{251}\right)\) | \(e\left(\frac{463}{502}\right)\) | \(e\left(\frac{33}{251}\right)\) | \(e\left(\frac{181}{502}\right)\) | \(e\left(\frac{389}{502}\right)\) | \(e\left(\frac{17}{502}\right)\) |
| \(\chi_{4024}(459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{150}{251}\right)\) | \(e\left(\frac{223}{251}\right)\) | \(e\left(\frac{455}{502}\right)\) | \(e\left(\frac{49}{251}\right)\) | \(e\left(\frac{79}{251}\right)\) | \(e\left(\frac{335}{502}\right)\) | \(e\left(\frac{122}{251}\right)\) | \(e\left(\frac{479}{502}\right)\) | \(e\left(\frac{31}{502}\right)\) | \(e\left(\frac{253}{502}\right)\) |
| \(\chi_{4024}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{126}{251}\right)\) | \(e\left(\frac{107}{251}\right)\) | \(e\left(\frac{81}{502}\right)\) | \(e\left(\frac{1}{251}\right)\) | \(e\left(\frac{227}{251}\right)\) | \(e\left(\frac{181}{502}\right)\) | \(e\left(\frac{233}{251}\right)\) | \(e\left(\frac{61}{502}\right)\) | \(e\left(\frac{267}{502}\right)\) | \(e\left(\frac{333}{502}\right)\) |