from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([250,250,431]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,10000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(5000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(500\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 5000.bt | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10000}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{500}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{159}{500}\right)\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{351}{500}\right)\) |
| \(\chi_{10000}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{349}{500}\right)\) |
| \(\chi_{10000}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{500}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{267}{500}\right)\) |
| \(\chi_{10000}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{500}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{77}{500}\right)\) | \(e\left(\frac{389}{500}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{453}{500}\right)\) |
| \(\chi_{10000}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{223}{500}\right)\) |
| \(\chi_{10000}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{500}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{173}{500}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{317}{500}\right)\) | \(e\left(\frac{297}{500}\right)\) |
| \(\chi_{10000}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{459}{500}\right)\) | \(e\left(\frac{419}{500}\right)\) |
| \(\chi_{10000}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{500}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{181}{500}\right)\) |
| \(\chi_{10000}(423,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{500}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{303}{500}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{291}{500}\right)\) | \(e\left(\frac{331}{500}\right)\) |
| \(\chi_{10000}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{497}{500}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{309}{500}\right)\) | \(e\left(\frac{269}{500}\right)\) |
| \(\chi_{10000}(503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{500}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{311}{500}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{467}{500}\right)\) | \(e\left(\frac{447}{500}\right)\) |
| \(\chi_{10000}(567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{500}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{449}{500}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{73}{500}\right)\) |
| \(\chi_{10000}(583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{500}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{339}{500}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{83}{500}\right)\) | \(e\left(\frac{103}{500}\right)\) |
| \(\chi_{10000}(647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{53}{500}\right)\) | \(e\left(\frac{21}{500}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{337}{500}\right)\) | \(e\left(\frac{117}{500}\right)\) |
| \(\chi_{10000}(663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{333}{500}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{339}{500}\right)\) | \(e\left(\frac{499}{500}\right)\) |
| \(\chi_{10000}(727,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{309}{500}\right)\) | \(e\left(\frac{113}{500}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{201}{500}\right)\) |
| \(\chi_{10000}(823,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{437}{500}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{299}{500}\right)\) | \(e\left(\frac{43}{500}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{71}{500}\right)\) | \(e\left(\frac{311}{500}\right)\) |
| \(\chi_{10000}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{500}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{201}{500}\right)\) | \(e\left(\frac{457}{500}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{429}{500}\right)\) | \(e\left(\frac{189}{500}\right)\) |
| \(\chi_{10000}(903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{500}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{151}{500}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{127}{500}\right)\) |
| \(\chi_{10000}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{237}{500}\right)\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{73}{500}\right)\) | \(e\left(\frac{193}{500}\right)\) |
| \(\chi_{10000}(983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{347}{500}\right)\) | \(e\left(\frac{279}{500}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{263}{500}\right)\) | \(e\left(\frac{483}{500}\right)\) |
| \(\chi_{10000}(1047,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{181}{500}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{357}{500}\right)\) | \(e\left(\frac{437}{500}\right)\) |
| \(\chi_{10000}(1063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{211}{500}\right)\) | \(e\left(\frac{27}{500}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{79}{500}\right)\) |
| \(\chi_{10000}(1127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{407}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{489}{500}\right)\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{81}{500}\right)\) | \(e\left(\frac{221}{500}\right)\) |
| \(\chi_{10000}(1223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{351}{500}\right)\) | \(e\left(\frac{291}{500}\right)\) |
| \(\chi_{10000}(1287,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{500}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{481}{500}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{49}{500}\right)\) | \(e\left(\frac{109}{500}\right)\) |
| \(\chi_{10000}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{263}{500}\right)\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{307}{500}\right)\) |
| \(\chi_{10000}(1367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{317}{500}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{393}{500}\right)\) | \(e\left(\frac{313}{500}\right)\) |
| \(\chi_{10000}(1383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{267}{500}\right)\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{443}{500}\right)\) | \(e\left(\frac{363}{500}\right)\) |
| \(\chi_{10000}(1447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{313}{500}\right)\) | \(e\left(\frac{341}{500}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{377}{500}\right)\) | \(e\left(\frac{257}{500}\right)\) |
| \(\chi_{10000}(1463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{99}{500}\right)\) | \(e\left(\frac{159}{500}\right)\) |