from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6275, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([75,174]))
chi.galois_orbit()
[g,chi] = znchar(Mod(89,6275))
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: | \(250\) | 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: | $\Q(\zeta_{125})$ |
| Fixed field: | Number field defined by a degree 250 polynomial (not computed) |
First 31 of 100 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6275}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{93}{250}\right)\) |
| \(\chi_{6275}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) |
| \(\chi_{6275}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{129}{250}\right)\) |
| \(\chi_{6275}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{179}{250}\right)\) |
| \(\chi_{6275}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{193}{250}\right)\) |
| \(\chi_{6275}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{151}{250}\right)\) |
| \(\chi_{6275}(344,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{39}{250}\right)\) |
| \(\chi_{6275}(469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{89}{250}\right)\) |
| \(\chi_{6275}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{211}{250}\right)\) |
| \(\chi_{6275}(644,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{69}{250}\right)\) |
| \(\chi_{6275}(654,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) |
| \(\chi_{6275}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{17}{250}\right)\) |
| \(\chi_{6275}(784,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{177}{250}\right)\) |
| \(\chi_{6275}(819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{109}{250}\right)\) |
| \(\chi_{6275}(884,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{207}{250}\right)\) |
| \(\chi_{6275}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{59}{250}\right)\) |
| \(\chi_{6275}(1089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{53}{250}\right)\) |
| \(\chi_{6275}(1159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{87}{250}\right)\) |
| \(\chi_{6275}(1179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{181}{250}\right)\) |
| \(\chi_{6275}(1339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{13}{250}\right)\) |
| \(\chi_{6275}(1464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{243}{250}\right)\) |
| \(\chi_{6275}(1469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{199}{250}\right)\) |
| \(\chi_{6275}(1509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) |
| \(\chi_{6275}(1519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) |
| \(\chi_{6275}(1589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{33}{250}\right)\) |
| \(\chi_{6275}(1659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{117}{250}\right)\) |
| \(\chi_{6275}(1679,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{31}{250}\right)\) |
| \(\chi_{6275}(1704,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{221}{250}\right)\) |
| \(\chi_{6275}(1879,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) |
| \(\chi_{6275}(1954,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{231}{250}\right)\) |
| \(\chi_{6275}(1979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{81}{250}\right)\) |