from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([0,46]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,2500))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2500\) | |
| Conductor: | \(625\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(125\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 625.j | 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 125 polynomial (not computed) |
First 31 of 100 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2500}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{8}{125}\right)\) |
| \(\chi_{2500}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{31}{125}\right)\) |
| \(\chi_{2500}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) |
| \(\chi_{2500}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) |
| \(\chi_{2500}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) |
| \(\chi_{2500}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{121}{125}\right)\) |
| \(\chi_{2500}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{9}{125}\right)\) |
| \(\chi_{2500}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) |
| \(\chi_{2500}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) |
| \(\chi_{2500}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{86}{125}\right)\) |
| \(\chi_{2500}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{49}{125}\right)\) |
| \(\chi_{2500}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) |
| \(\chi_{2500}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{53}{125}\right)\) |
| \(\chi_{2500}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{51}{125}\right)\) |
| \(\chi_{2500}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) |
| \(\chi_{2500}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) |
| \(\chi_{2500}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{68}{125}\right)\) |
| \(\chi_{2500}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) |
| \(\chi_{2500}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{4}{125}\right)\) |
| \(\chi_{2500}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) |
| \(\chi_{2500}(521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) |
| \(\chi_{2500}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) |
| \(\chi_{2500}(561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) |
| \(\chi_{2500}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{47}{125}\right)\) |
| \(\chi_{2500}(621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{98}{125}\right)\) |
| \(\chi_{2500}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{71}{125}\right)\) |
| \(\chi_{2500}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{84}{125}\right)\) |
| \(\chi_{2500}(681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{37}{125}\right)\) |
| \(\chi_{2500}(721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) |
| \(\chi_{2500}(741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{36}{125}\right)\) |
| \(\chi_{2500}(761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{124}{125}\right)\) |