from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6275, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([425,474]))
chi.galois_orbit()
[g,chi] = znchar(Mod(72,6275))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6275\) | |
| Conductor: | \(6275\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6275}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{59}{500}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{177}{500}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{343}{500}\right)\) |
| \(\chi_{6275}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{229}{500}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{311}{500}\right)\) |
| \(\chi_{6275}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{317}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{451}{500}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{309}{500}\right)\) |
| \(\chi_{6275}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{167}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{1}{500}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{357}{500}\right)\) | \(e\left(\frac{259}{500}\right)\) |
| \(\chi_{6275}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{323}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{469}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{471}{500}\right)\) |
| \(\chi_{6275}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{429}{500}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{153}{500}\right)\) | \(e\left(\frac{111}{500}\right)\) |
| \(\chi_{6275}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{257}{500}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{63}{500}\right)\) |
| \(\chi_{6275}(202,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{337}{500}\right)\) | \(e\left(\frac{19}{500}\right)\) |
| \(\chi_{6275}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{437}{500}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{311}{500}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{27}{500}\right)\) | \(e\left(\frac{49}{500}\right)\) |
| \(\chi_{6275}(322,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{123}{500}\right)\) |
| \(\chi_{6275}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{493}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{3}{500}\right)\) | \(e\left(\frac{61}{500}\right)\) |
| \(\chi_{6275}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{413}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{467}{500}\right)\) |
| \(\chi_{6275}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{3}{500}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{331}{500}\right)\) |
| \(\chi_{6275}(388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{73}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{183}{500}\right)\) | \(e\left(\frac{221}{500}\right)\) |
| \(\chi_{6275}(413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{393}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{361}{500}\right)\) |
| \(\chi_{6275}(427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{21}{500}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{61}{250}\right)\) | \(e\left(\frac{497}{500}\right)\) | \(e\left(\frac{439}{500}\right)\) |
| \(\chi_{6275}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{369}{500}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{213}{500}\right)\) |
| \(\chi_{6275}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{113}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{341}{500}\right)\) | \(e\left(\frac{267}{500}\right)\) |
| \(\chi_{6275}(528,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{113}{500}\right)\) |
| \(\chi_{6275}(558,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{21}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{317}{500}\right)\) |
| \(\chi_{6275}(598,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{111}{500}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{127}{500}\right)\) | \(e\left(\frac{249}{500}\right)\) |
| \(\chi_{6275}(638,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{259}{500}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{463}{500}\right)\) | \(e\left(\frac{81}{500}\right)\) |
| \(\chi_{6275}(652,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{27}{500}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{81}{500}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{479}{500}\right)\) |
| \(\chi_{6275}(712,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{203}{500}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{109}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{413}{500}\right)\) | \(e\left(\frac{231}{500}\right)\) |
| \(\chi_{6275}(767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{487}{500}\right)\) |
| \(\chi_{6275}(777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{247}{500}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{419}{500}\right)\) |
| \(\chi_{6275}(787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{463}{500}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{389}{500}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{251}{500}\right)\) |
| \(\chi_{6275}(873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{147}{500}\right)\) | \(e\left(\frac{489}{500}\right)\) |
| \(\chi_{6275}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{33}{500}\right)\) | \(e\left(\frac{171}{500}\right)\) |
| \(\chi_{6275}(892,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{291}{500}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{107}{500}\right)\) |
| \(\chi_{6275}(912,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{383}{500}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{149}{500}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{91}{500}\right)\) |