from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([250,107]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,2500))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2500\) | |
| Conductor: | \(2500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(500\) | 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: | $\Q(\zeta_{500})$ |
| Fixed field: | Number field defined by a degree 500 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2500}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{367}{500}\right)\) | \(e\left(\frac{97}{500}\right)\) |
| \(\chi_{2500}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{500}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{409}{500}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{101}{500}\right)\) |
| \(\chi_{2500}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{33}{500}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{291}{500}\right)\) |
| \(\chi_{2500}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{387}{500}\right)\) |
| \(\chi_{2500}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{500}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{19}{500}\right)\) | \(e\left(\frac{129}{500}\right)\) |
| \(\chi_{2500}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{307}{500}\right)\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{423}{500}\right)\) |
| \(\chi_{2500}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{500}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{277}{500}\right)\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{253}{500}\right)\) |
| \(\chi_{2500}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{500}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{391}{500}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{99}{500}\right)\) |
| \(\chi_{2500}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{339}{500}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{153}{500}\right)\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{17}{500}\right)\) |
| \(\chi_{2500}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{407}{500}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{489}{500}\right)\) | \(e\left(\frac{123}{500}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{331}{500}\right)\) | \(e\left(\frac{221}{500}\right)\) |
| \(\chi_{2500}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{171}{500}\right)\) |
| \(\chi_{2500}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{489}{500}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{203}{500}\right)\) | \(e\left(\frac{321}{500}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{437}{500}\right)\) | \(e\left(\frac{467}{500}\right)\) |
| \(\chi_{2500}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{500}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{341}{500}\right)\) | \(e\left(\frac{187}{500}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{239}{500}\right)\) | \(e\left(\frac{149}{500}\right)\) |
| \(\chi_{2500}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{327}{500}\right)\) | \(e\left(\frac{389}{500}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{203}{500}\right)\) |
| \(\chi_{2500}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{257}{500}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{473}{500}\right)\) |
| \(\chi_{2500}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{211}{500}\right)\) | \(e\left(\frac{277}{500}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{469}{500}\right)\) | \(e\left(\frac{79}{500}\right)\) |
| \(\chi_{2500}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{437}{500}\right)\) |
| \(\chi_{2500}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{183}{500}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{151}{500}\right)\) | \(e\left(\frac{341}{500}\right)\) |
| \(\chi_{2500}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{500}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{459}{500}\right)\) | \(e\left(\frac{413}{500}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{461}{500}\right)\) | \(e\left(\frac{51}{500}\right)\) |
| \(\chi_{2500}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{500}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{317}{500}\right)\) | \(e\left(\frac{47}{500}\right)\) |
| \(\chi_{2500}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{21}{500}\right)\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{459}{500}\right)\) | \(e\left(\frac{169}{500}\right)\) |
| \(\chi_{2500}(267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{347}{500}\right)\) | \(e\left(\frac{29}{500}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{13}{500}\right)\) | \(e\left(\frac{483}{500}\right)\) |
| \(\chi_{2500}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{237}{500}\right)\) | \(e\left(\frac{259}{500}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{323}{500}\right)\) | \(e\left(\frac{193}{500}\right)\) |
| \(\chi_{2500}(287,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{31}{500}\right)\) | \(e\left(\frac{17}{500}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{59}{500}\right)\) |
| \(\chi_{2500}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{391}{500}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{227}{500}\right)\) | \(e\left(\frac{357}{500}\right)\) |
| \(\chi_{2500}(323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{149}{500}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{461}{500}\right)\) |
| \(\chi_{2500}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{477}{500}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{379}{500}\right)\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{431}{500}\right)\) |
| \(\chi_{2500}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{500}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{197}{500}\right)\) | \(e\left(\frac{127}{500}\right)\) |
| \(\chi_{2500}(363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{201}{500}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{189}{500}\right)\) |
| \(\chi_{2500}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{367}{500}\right)\) | \(e\left(\frac{169}{500}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{93}{500}\right)\) | \(e\left(\frac{263}{500}\right)\) |
| \(\chi_{2500}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{217}{500}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{413}{500}\right)\) |