from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([0,125,92]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,10000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(10000\) | |
| Conductor: | \(10000\) | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10000}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{500}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{163}{500}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{331}{500}\right)\) | \(e\left(\frac{89}{500}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{157}{500}\right)\) |
| \(\chi_{10000}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{233}{500}\right)\) | \(e\left(\frac{327}{500}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{251}{500}\right)\) |
| \(\chi_{10000}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{453}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{481}{500}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{397}{500}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{359}{500}\right)\) |
| \(\chi_{10000}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{491}{500}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{113}{500}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{459}{500}\right)\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{473}{500}\right)\) |
| \(\chi_{10000}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{387}{500}\right)\) | \(e\left(\frac{493}{500}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{41}{500}\right)\) | \(e\left(\frac{79}{500}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{27}{500}\right)\) |
| \(\chi_{10000}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{139}{500}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{43}{500}\right)\) | \(e\left(\frac{217}{500}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{321}{500}\right)\) |
| \(\chi_{10000}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{443}{500}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{49}{500}\right)\) | \(e\left(\frac{211}{500}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{407}{500}\right)\) | \(e\left(\frac{333}{500}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{329}{500}\right)\) |
| \(\chi_{10000}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{383}{500}\right)\) | \(e\left(\frac{37}{500}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{143}{500}\right)\) |
| \(\chi_{10000}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{500}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{357}{500}\right)\) | \(e\left(\frac{323}{500}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{251}{500}\right)\) | \(e\left(\frac{69}{500}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{397}{500}\right)\) |
| \(\chi_{10000}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{297}{500}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{353}{500}\right)\) | \(e\left(\frac{107}{500}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{391}{500}\right)\) |
| \(\chi_{10000}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{299}{500}\right)\) |
| \(\chi_{10000}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{153}{500}\right)\) | \(e\left(\frac{67}{500}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{101}{500}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{313}{500}\right)\) |
| \(\chi_{10000}(621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{327}{500}\right)\) | \(e\left(\frac{153}{500}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{461}{500}\right)\) | \(e\left(\frac{59}{500}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{267}{500}\right)\) |
| \(\chi_{10000}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{163}{500}\right)\) | \(e\left(\frac{497}{500}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{461}{500}\right)\) |
| \(\chi_{10000}(741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{171}{500}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{269}{500}\right)\) |
| \(\chi_{10000}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{500}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{289}{500}\right)\) | \(e\left(\frac{191}{500}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{483}{500}\right)\) |
| \(\chi_{10000}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{379}{500}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{297}{500}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{171}{500}\right)\) | \(e\left(\frac{49}{500}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{137}{500}\right)\) |
| \(\chi_{10000}(861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{500}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{111}{500}\right)\) | \(e\left(\frac{29}{500}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{387}{500}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{31}{500}\right)\) |
| \(\chi_{10000}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{413}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{259}{500}\right)\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{437}{500}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{239}{500}\right)\) |
| \(\chi_{10000}(981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{500}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{127}{500}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{153}{500}\right)\) |
| \(\chi_{10000}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{500}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{267}{500}\right)\) | \(e\left(\frac{313}{500}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{7}{500}\right)\) |
| \(\chi_{10000}(1061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{500}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{281}{500}\right)\) | \(e\left(\frac{159}{500}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{277}{500}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{101}{500}\right)\) |
| \(\chi_{10000}(1141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{329}{500}\right)\) | \(e\left(\frac{131}{500}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{447}{500}\right)\) | \(e\left(\frac{93}{500}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{209}{500}\right)\) |
| \(\chi_{10000}(1181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{441}{500}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{463}{500}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{9}{500}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{323}{500}\right)\) |
| \(\chi_{10000}(1221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{459}{500}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{237}{500}\right)\) | \(e\left(\frac{143}{500}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{91}{500}\right)\) | \(e\left(\frac{29}{500}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{377}{500}\right)\) |
| \(\chi_{10000}(1261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{451}{500}\right)\) | \(e\left(\frac{289}{500}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{93}{500}\right)\) | \(e\left(\frac{167}{500}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{171}{500}\right)\) |
| \(\chi_{10000}(1341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{393}{500}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{399}{500}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{457}{500}\right)\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{179}{500}\right)\) |
| \(\chi_{10000}(1381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{500}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{233}{500}\right)\) | \(e\left(\frac{187}{500}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{119}{500}\right)\) | \(e\left(\frac{461}{500}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{493}{500}\right)\) |
| \(\chi_{10000}(1421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{301}{500}\right)\) | \(e\left(\frac{19}{500}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{247}{500}\right)\) |
| \(\chi_{10000}(1461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{500}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{121}{500}\right)\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{403}{500}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{241}{500}\right)\) |
| \(\chi_{10000}(1541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{500}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{469}{500}\right)\) | \(e\left(\frac{91}{500}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{467}{500}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{149}{500}\right)\) |