from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([0,125,22]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,10000))
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: | \(250\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 5000.bn | 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: | $\Q(\zeta_{125})$ |
| Fixed field: | Number field defined by a degree 250 polynomial (not computed) |
First 31 of 100 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10000}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{187}{250}\right)\) |
| \(\chi_{10000}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{171}{250}\right)\) |
| \(\chi_{10000}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{29}{250}\right)\) |
| \(\chi_{10000}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{53}{250}\right)\) |
| \(\chi_{10000}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{157}{250}\right)\) |
| \(\chi_{10000}(521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{41}{250}\right)\) |
| \(\chi_{10000}(681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{199}{250}\right)\) |
| \(\chi_{10000}(761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{123}{250}\right)\) |
| \(\chi_{10000}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{127}{250}\right)\) |
| \(\chi_{10000}(921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{161}{250}\right)\) |
| \(\chi_{10000}(1081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{119}{250}\right)\) |
| \(\chi_{10000}(1161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{193}{250}\right)\) |
| \(\chi_{10000}(1241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{97}{250}\right)\) |
| \(\chi_{10000}(1321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{31}{250}\right)\) |
| \(\chi_{10000}(1481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{39}{250}\right)\) |
| \(\chi_{10000}(1561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) |
| \(\chi_{10000}(1641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{67}{250}\right)\) |
| \(\chi_{10000}(1721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{151}{250}\right)\) |
| \(\chi_{10000}(1881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) |
| \(\chi_{10000}(1961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{83}{250}\right)\) |
| \(\chi_{10000}(2041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{37}{250}\right)\) |
| \(\chi_{10000}(2121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{21}{250}\right)\) |
| \(\chi_{10000}(2281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{129}{250}\right)\) |
| \(\chi_{10000}(2361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{153}{250}\right)\) |
| \(\chi_{10000}(2441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{7}{250}\right)\) |
| \(\chi_{10000}(2521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{141}{250}\right)\) |
| \(\chi_{10000}(2681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{49}{250}\right)\) |
| \(\chi_{10000}(2761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{223}{250}\right)\) |
| \(\chi_{10000}(2841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{227}{250}\right)\) |
| \(\chi_{10000}(2921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{11}{250}\right)\) |
| \(\chi_{10000}(3081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{219}{250}\right)\) |