from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6002, base_ring=CyclotomicField(750))
M = H._module
chi = DirichletCharacter(H, M([362]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,6002))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6002\) | |
| Conductor: | \(3001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(375\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3001.z | 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: | $\Q(\zeta_{375})$ |
| Fixed field: | Number field defined by a degree 375 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6002}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{338}{375}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{88}{125}\right)\) |
| \(\chi_{6002}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{277}{375}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{27}{125}\right)\) |
| \(\chi_{6002}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{22}{375}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{22}{125}\right)\) |
| \(\chi_{6002}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{233}{375}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{108}{125}\right)\) |
| \(\chi_{6002}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{139}{375}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{14}{125}\right)\) |
| \(\chi_{6002}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{301}{375}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{51}{125}\right)\) |
| \(\chi_{6002}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{154}{375}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{29}{125}\right)\) |
| \(\chi_{6002}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{8}{375}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{8}{125}\right)\) |
| \(\chi_{6002}(309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{371}{375}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{121}{125}\right)\) |
| \(\chi_{6002}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{172}{375}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{47}{125}\right)\) |
| \(\chi_{6002}(425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{122}{375}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{122}{125}\right)\) |
| \(\chi_{6002}(469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{217}{375}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{92}{125}\right)\) |
| \(\chi_{6002}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{251}{375}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{1}{125}\right)\) |
| \(\chi_{6002}(531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{361}{375}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{111}{125}\right)\) |
| \(\chi_{6002}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{244}{375}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{119}{125}\right)\) |
| \(\chi_{6002}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{349}{375}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{99}{125}\right)\) |
| \(\chi_{6002}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{41}{375}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{41}{125}\right)\) |
| \(\chi_{6002}(609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{311}{375}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{61}{125}\right)\) |
| \(\chi_{6002}(629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{73}{375}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{73}{125}\right)\) |
| \(\chi_{6002}(633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{284}{375}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{34}{125}\right)\) |
| \(\chi_{6002}(665,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{239}{375}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{114}{125}\right)\) |
| \(\chi_{6002}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{206}{375}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{81}{125}\right)\) |
| \(\chi_{6002}(739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{58}{375}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{58}{125}\right)\) |
| \(\chi_{6002}(765,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{329}{375}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{79}{125}\right)\) |
| \(\chi_{6002}(775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{61}{375}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{61}{125}\right)\) |
| \(\chi_{6002}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{236}{375}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{111}{125}\right)\) |
| \(\chi_{6002}(805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{148}{375}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{23}{125}\right)\) |
| \(\chi_{6002}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{173}{375}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{48}{125}\right)\) |
| \(\chi_{6002}(897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{86}{375}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{86}{125}\right)\) |
| \(\chi_{6002}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{104}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{343}{375}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{93}{125}\right)\) |
| \(\chi_{6002}(925,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{269}{375}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{19}{125}\right)\) |