from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([0,69,266]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,6815))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(6815\) | |
Conductor: | \(1363\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(322\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 1363.u | 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: | $\Q(\zeta_{161})$ |
Fixed field: | Number field defined by a degree 322 polynomial (not computed) |
First 31 of 132 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{6815}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{45}{322}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{152}{161}\right)\) |
\(\chi_{6815}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{52}{161}\right)\) |
\(\chi_{6815}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{32}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{43}{161}\right)\) |
\(\chi_{6815}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{65}{322}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{118}{161}\right)\) |
\(\chi_{6815}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{45}{161}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{79}{161}\right)\) |
\(\chi_{6815}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{101}{161}\right)\) |
\(\chi_{6815}(216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{134}{161}\right)\) |
\(\chi_{6815}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{86}{161}\right)\) | \(e\left(\frac{116}{161}\right)\) | \(e\left(\frac{219}{322}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{83}{161}\right)\) |
\(\chi_{6815}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{261}{322}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{100}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{131}{161}\right)\) |
\(\chi_{6815}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{110}{161}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{17}{161}\right)\) |
\(\chi_{6815}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{116}{161}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{111}{161}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{15}{161}\right)\) |
\(\chi_{6815}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{32}{161}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{117}{161}\right)\) |
\(\chi_{6815}(486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{150}{161}\right)\) |
\(\chi_{6815}(506,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{123}{161}\right)\) | \(e\left(\frac{103}{161}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{120}{161}\right)\) |
\(\chi_{6815}(526,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{110}{161}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{137}{161}\right)\) |
\(\chi_{6815}(531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{62}{161}\right)\) |
\(\chi_{6815}(676,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{102}{161}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{48}{161}\right)\) |
\(\chi_{6815}(776,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{100}{161}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{102}{161}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{66}{161}\right)\) |
\(\chi_{6815}(816,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{18}{161}\right)\) |
\(\chi_{6815}(921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{67}{161}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{157}{161}\right)\) |
\(\chi_{6815}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{150}{161}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{116}{161}\right)\) |
\(\chi_{6815}(991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{121}{161}\right)\) |
\(\chi_{6815}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{159}{161}\right)\) |
\(\chi_{6815}(1066,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{38}{161}\right)\) |
\(\chi_{6815}(1106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{13}{161}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{123}{161}\right)\) |
\(\chi_{6815}(1136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{9}{161}\right)\) |
\(\chi_{6815}(1211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{143}{161}\right)\) |
\(\chi_{6815}(1231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{22}{161}\right)\) |
\(\chi_{6815}(1256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{22}{161}\right)\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{90}{161}\right)\) |
\(\chi_{6815}(1281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{86}{161}\right)\) |
\(\chi_{6815}(1311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{163}{322}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{43}{161}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{96}{161}\right)\) |