from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([250,0,97]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,10000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(2500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(500\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2500.x | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10000}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{387}{500}\right)\) |
| \(\chi_{10000}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{500}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{19}{500}\right)\) | \(e\left(\frac{129}{500}\right)\) |
| \(\chi_{10000}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{171}{500}\right)\) |
| \(\chi_{10000}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{183}{500}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{151}{500}\right)\) | \(e\left(\frac{341}{500}\right)\) |
| \(\chi_{10000}(287,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{31}{500}\right)\) | \(e\left(\frac{17}{500}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{59}{500}\right)\) |
| \(\chi_{10000}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{391}{500}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{227}{500}\right)\) | \(e\left(\frac{357}{500}\right)\) |
| \(\chi_{10000}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{367}{500}\right)\) | \(e\left(\frac{169}{500}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{93}{500}\right)\) | \(e\left(\frac{263}{500}\right)\) |
| \(\chi_{10000}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{471}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{217}{500}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{413}{500}\right)\) |
| \(\chi_{10000}(447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{363}{500}\right)\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{77}{500}\right)\) | \(e\left(\frac{207}{500}\right)\) |
| \(\chi_{10000}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{467}{500}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{209}{500}\right)\) |
| \(\chi_{10000}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{500}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{233}{500}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{1}{500}\right)\) | \(e\left(\frac{191}{500}\right)\) |
| \(\chi_{10000}(623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{389}{500}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{321}{500}\right)\) |
| \(\chi_{10000}(687,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{493}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{311}{500}\right)\) | \(e\left(\frac{477}{500}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{369}{500}\right)\) | \(e\left(\frac{479}{500}\right)\) |
| \(\chi_{10000}(703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{333}{500}\right)\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{37}{500}\right)\) |
| \(\chi_{10000}(767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{461}{500}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{229}{500}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{413}{500}\right)\) | \(e\left(\frac{383}{500}\right)\) |
| \(\chi_{10000}(783,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{137}{500}\right)\) | \(e\left(\frac{59}{500}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{293}{500}\right)\) |
| \(\chi_{10000}(847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{27}{500}\right)\) |
| \(\chi_{10000}(863,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{500}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{279}{500}\right)\) | \(e\left(\frac{289}{500}\right)\) |
| \(\chi_{10000}(927,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{237}{500}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{493}{500}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{221}{500}\right)\) | \(e\left(\frac{211}{500}\right)\) |
| \(\chi_{10000}(1023,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{267}{500}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{209}{500}\right)\) | \(e\left(\frac{163}{500}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{211}{500}\right)\) | \(e\left(\frac{301}{500}\right)\) |
| \(\chi_{10000}(1087,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{500}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{91}{500}\right)\) | \(e\left(\frac{437}{500}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{489}{500}\right)\) | \(e\left(\frac{399}{500}\right)\) |
| \(\chi_{10000}(1103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{500}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{453}{500}\right)\) | \(e\left(\frac{71}{500}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{187}{500}\right)\) | \(e\left(\frac{217}{500}\right)\) |
| \(\chi_{10000}(1167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{500}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{27}{500}\right)\) | \(e\left(\frac{289}{500}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{233}{500}\right)\) | \(e\left(\frac{3}{500}\right)\) |
| \(\chi_{10000}(1183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{500}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{499}{500}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{103}{500}\right)\) | \(e\left(\frac{173}{500}\right)\) |
| \(\chi_{10000}(1247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{500}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{123}{500}\right)\) | \(e\left(\frac{261}{500}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{117}{500}\right)\) | \(e\left(\frac{347}{500}\right)\) |
| \(\chi_{10000}(1263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{321}{500}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{159}{500}\right)\) | \(e\left(\frac{369}{500}\right)\) |
| \(\chi_{10000}(1327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{500}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{79}{500}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{231}{500}\right)\) |
| \(\chi_{10000}(1423,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{29}{500}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{281}{500}\right)\) |
| \(\chi_{10000}(1487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{273}{500}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{397}{500}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{109}{500}\right)\) | \(e\left(\frac{319}{500}\right)\) |
| \(\chi_{10000}(1503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{500}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{73}{500}\right)\) | \(e\left(\frac{411}{500}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{167}{500}\right)\) | \(e\left(\frac{397}{500}\right)\) |
| \(\chi_{10000}(1567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{500}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{349}{500}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{53}{500}\right)\) | \(e\left(\frac{123}{500}\right)\) |