from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([0,139]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,2500))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2500\) | |
| Conductor: | \(625\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(500\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 625.l | 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: | $\Q(\zeta_{500})$ |
| Fixed field: | Number field defined by a degree 500 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2500}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{321}{500}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{409}{500}\right)\) | \(e\left(\frac{119}{500}\right)\) |
| \(\chi_{2500}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{429}{500}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{33}{500}\right)\) |
| \(\chi_{2500}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{359}{500}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{273}{500}\right)\) | \(e\left(\frac{143}{500}\right)\) |
| \(\chi_{2500}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{299}{500}\right)\) | \(e\left(\frac{109}{500}\right)\) |
| \(\chi_{2500}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{13}{500}\right)\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{177}{500}\right)\) | \(e\left(\frac{307}{500}\right)\) |
| \(\chi_{2500}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{500}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{449}{500}\right)\) | \(e\left(\frac{343}{500}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{411}{500}\right)\) |
| \(\chi_{2500}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{327}{500}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{79}{500}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{191}{500}\right)\) | \(e\left(\frac{481}{500}\right)\) |
| \(\chi_{2500}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{500}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{343}{500}\right)\) | \(e\left(\frac{301}{500}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{247}{500}\right)\) | \(e\left(\frac{177}{500}\right)\) |
| \(\chi_{2500}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{1}{500}\right)\) | \(e\left(\frac{307}{500}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{139}{500}\right)\) |
| \(\chi_{2500}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{313}{500}\right)\) |
| \(\chi_{2500}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{17}{500}\right)\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{363}{500}\right)\) |
| \(\chi_{2500}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{363}{500}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{51}{500}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{79}{500}\right)\) | \(e\left(\frac{89}{500}\right)\) |
| \(\chi_{2500}(153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{500}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{293}{500}\right)\) | \(e\left(\frac{451}{500}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{297}{500}\right)\) | \(e\left(\frac{227}{500}\right)\) |
| \(\chi_{2500}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{500}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{29}{500}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{31}{500}\right)\) |
| \(\chi_{2500}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{500}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{499}{500}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{361}{500}\right)\) |
| \(\chi_{2500}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{341}{500}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{127}{500}\right)\) | \(e\left(\frac{257}{500}\right)\) |
| \(\chi_{2500}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{500}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{181}{500}\right)\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{349}{500}\right)\) | \(e\left(\frac{159}{500}\right)\) |
| \(\chi_{2500}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{500}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{87}{500}\right)\) | \(e\left(\frac{209}{500}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{223}{500}\right)\) | \(e\left(\frac{93}{500}\right)\) |
| \(\chi_{2500}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{497}{500}\right)\) | \(e\left(\frac{79}{500}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{113}{500}\right)\) | \(e\left(\frac{83}{500}\right)\) |
| \(\chi_{2500}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{397}{500}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{359}{500}\right)\) | \(e\left(\frac{69}{500}\right)\) |
| \(\chi_{2500}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{500}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{73}{500}\right)\) | \(e\left(\frac{411}{500}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{147}{500}\right)\) |
| \(\chi_{2500}(273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{500}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{109}{500}\right)\) | \(e\left(\frac{463}{500}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{61}{500}\right)\) | \(e\left(\frac{151}{500}\right)\) |
| \(\chi_{2500}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{133}{500}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{51}{500}\right)\) | \(e\left(\frac{241}{500}\right)\) |
| \(\chi_{2500}(297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{279}{500}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{337}{500}\right)\) |
| \(\chi_{2500}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{393}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{327}{500}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{179}{500}\right)\) |
| \(\chi_{2500}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{291}{500}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{349}{500}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{303}{500}\right)\) | \(e\left(\frac{373}{500}\right)\) |
| \(\chi_{2500}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{477}{500}\right)\) | \(e\left(\frac{439}{500}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{33}{500}\right)\) | \(e\left(\frac{303}{500}\right)\) |
| \(\chi_{2500}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{500}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{191}{500}\right)\) | \(e\left(\frac{137}{500}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{139}{500}\right)\) | \(e\left(\frac{49}{500}\right)\) |
| \(\chi_{2500}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{67}{500}\right)\) |
| \(\chi_{2500}(373,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{271}{500}\right)\) |
| \(\chi_{2500}(377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{339}{500}\right)\) | \(e\left(\frac{73}{500}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{121}{500}\right)\) |