from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(788))
M = H._module
chi = DirichletCharacter(H, M([53]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,4729))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4729\) | |
| Conductor: | \(4729\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(788\) | 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: | $\Q(\zeta_{788})$ |
| Fixed field: | Number field defined by a degree 788 polynomial (not computed) |
First 31 of 392 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4729}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{394}\right)\) | \(e\left(\frac{171}{394}\right)\) | \(e\left(\frac{153}{197}\right)\) | \(e\left(\frac{87}{394}\right)\) | \(e\left(\frac{162}{197}\right)\) | \(e\left(\frac{13}{394}\right)\) | \(e\left(\frac{65}{394}\right)\) | \(e\left(\frac{171}{197}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{143}{788}\right)\) |
| \(\chi_{4729}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{394}\right)\) | \(e\left(\frac{307}{394}\right)\) | \(e\left(\frac{171}{197}\right)\) | \(e\left(\frac{329}{394}\right)\) | \(e\left(\frac{42}{197}\right)\) | \(e\left(\frac{339}{394}\right)\) | \(e\left(\frac{119}{394}\right)\) | \(e\left(\frac{110}{197}\right)\) | \(e\left(\frac{53}{197}\right)\) | \(e\left(\frac{183}{788}\right)\) |
| \(\chi_{4729}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{394}\right)\) | \(e\left(\frac{119}{394}\right)\) | \(e\left(\frac{65}{197}\right)\) | \(e\left(\frac{261}{394}\right)\) | \(e\left(\frac{92}{197}\right)\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{195}{394}\right)\) | \(e\left(\frac{119}{197}\right)\) | \(e\left(\frac{163}{197}\right)\) | \(e\left(\frac{429}{788}\right)\) |
| \(\chi_{4729}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{394}\right)\) | \(e\left(\frac{255}{394}\right)\) | \(e\left(\frac{83}{197}\right)\) | \(e\left(\frac{109}{394}\right)\) | \(e\left(\frac{169}{197}\right)\) | \(e\left(\frac{365}{394}\right)\) | \(e\left(\frac{249}{394}\right)\) | \(e\left(\frac{58}{197}\right)\) | \(e\left(\frac{96}{197}\right)\) | \(e\left(\frac{469}{788}\right)\) |
| \(\chi_{4729}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{394}\right)\) | \(e\left(\frac{391}{394}\right)\) | \(e\left(\frac{101}{197}\right)\) | \(e\left(\frac{351}{394}\right)\) | \(e\left(\frac{49}{197}\right)\) | \(e\left(\frac{297}{394}\right)\) | \(e\left(\frac{303}{394}\right)\) | \(e\left(\frac{194}{197}\right)\) | \(e\left(\frac{29}{197}\right)\) | \(e\left(\frac{509}{788}\right)\) |
| \(\chi_{4729}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{394}\right)\) | \(e\left(\frac{133}{394}\right)\) | \(e\left(\frac{119}{197}\right)\) | \(e\left(\frac{199}{394}\right)\) | \(e\left(\frac{126}{197}\right)\) | \(e\left(\frac{229}{394}\right)\) | \(e\left(\frac{357}{394}\right)\) | \(e\left(\frac{133}{197}\right)\) | \(e\left(\frac{159}{197}\right)\) | \(e\left(\frac{549}{788}\right)\) |
| \(\chi_{4729}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{371}{394}\right)\) | \(e\left(\frac{67}{394}\right)\) | \(e\left(\frac{174}{197}\right)\) | \(e\left(\frac{41}{394}\right)\) | \(e\left(\frac{22}{197}\right)\) | \(e\left(\frac{65}{394}\right)\) | \(e\left(\frac{325}{394}\right)\) | \(e\left(\frac{67}{197}\right)\) | \(e\left(\frac{9}{197}\right)\) | \(e\left(\frac{715}{788}\right)\) |
| \(\chi_{4729}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{394}\right)\) | \(e\left(\frac{203}{394}\right)\) | \(e\left(\frac{192}{197}\right)\) | \(e\left(\frac{283}{394}\right)\) | \(e\left(\frac{99}{197}\right)\) | \(e\left(\frac{391}{394}\right)\) | \(e\left(\frac{379}{394}\right)\) | \(e\left(\frac{6}{197}\right)\) | \(e\left(\frac{139}{197}\right)\) | \(e\left(\frac{755}{788}\right)\) |
| \(\chi_{4729}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{394}\right)\) | \(e\left(\frac{339}{394}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{131}{394}\right)\) | \(e\left(\frac{176}{197}\right)\) | \(e\left(\frac{323}{394}\right)\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{142}{197}\right)\) | \(e\left(\frac{72}{197}\right)\) | \(e\left(\frac{7}{788}\right)\) |
| \(\chi_{4729}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{394}\right)\) | \(e\left(\frac{81}{394}\right)\) | \(e\left(\frac{31}{197}\right)\) | \(e\left(\frac{373}{394}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{255}{394}\right)\) | \(e\left(\frac{93}{394}\right)\) | \(e\left(\frac{81}{197}\right)\) | \(e\left(\frac{5}{197}\right)\) | \(e\left(\frac{47}{788}\right)\) |
| \(\chi_{4729}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{394}\right)\) | \(e\left(\frac{183}{394}\right)\) | \(e\left(\frac{143}{197}\right)\) | \(e\left(\frac{259}{394}\right)\) | \(e\left(\frac{163}{197}\right)\) | \(e\left(\frac{7}{394}\right)\) | \(e\left(\frac{35}{394}\right)\) | \(e\left(\frac{183}{197}\right)\) | \(e\left(\frac{4}{197}\right)\) | \(e\left(\frac{471}{788}\right)\) |
| \(\chi_{4729}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{261}{394}\right)\) | \(e\left(\frac{199}{394}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{357}{394}\right)\) | \(e\left(\frac{33}{197}\right)\) | \(e\left(\frac{393}{394}\right)\) | \(e\left(\frac{389}{394}\right)\) | \(e\left(\frac{2}{197}\right)\) | \(e\left(\frac{112}{197}\right)\) | \(e\left(\frac{777}{788}\right)\) |
| \(\chi_{4729}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{394}\right)\) | \(e\left(\frac{89}{394}\right)\) | \(e\left(\frac{90}{197}\right)\) | \(e\left(\frac{225}{394}\right)\) | \(e\left(\frac{188}{197}\right)\) | \(e\left(\frac{251}{394}\right)\) | \(e\left(\frac{73}{394}\right)\) | \(e\left(\frac{89}{197}\right)\) | \(e\left(\frac{59}{197}\right)\) | \(e\left(\frac{3}{788}\right)\) |
| \(\chi_{4729}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{394}\right)\) | \(e\left(\frac{15}{394}\right)\) | \(e\left(\frac{86}{197}\right)\) | \(e\left(\frac{215}{394}\right)\) | \(e\left(\frac{149}{197}\right)\) | \(e\left(\frac{91}{394}\right)\) | \(e\left(\frac{61}{394}\right)\) | \(e\left(\frac{15}{197}\right)\) | \(e\left(\frac{52}{197}\right)\) | \(e\left(\frac{213}{788}\right)\) |
| \(\chi_{4729}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{394}\right)\) | \(e\left(\frac{257}{394}\right)\) | \(e\left(\frac{147}{197}\right)\) | \(e\left(\frac{269}{394}\right)\) | \(e\left(\frac{5}{197}\right)\) | \(e\left(\frac{167}{394}\right)\) | \(e\left(\frac{47}{394}\right)\) | \(e\left(\frac{60}{197}\right)\) | \(e\left(\frac{11}{197}\right)\) | \(e\left(\frac{261}{788}\right)\) |
| \(\chi_{4729}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{394}\right)\) | \(e\left(\frac{217}{394}\right)\) | \(e\left(\frac{49}{197}\right)\) | \(e\left(\frac{221}{394}\right)\) | \(e\left(\frac{133}{197}\right)\) | \(e\left(\frac{187}{394}\right)\) | \(e\left(\frac{147}{394}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{135}{197}\right)\) | \(e\left(\frac{87}{788}\right)\) |
| \(\chi_{4729}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{394}\right)\) | \(e\left(\frac{209}{394}\right)\) | \(e\left(\frac{187}{197}\right)\) | \(e\left(\frac{369}{394}\right)\) | \(e\left(\frac{1}{197}\right)\) | \(e\left(\frac{191}{394}\right)\) | \(e\left(\frac{167}{394}\right)\) | \(e\left(\frac{12}{197}\right)\) | \(e\left(\frac{81}{197}\right)\) | \(e\left(\frac{131}{788}\right)\) |
| \(\chi_{4729}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{301}{394}\right)\) | \(e\left(\frac{151}{394}\right)\) | \(e\left(\frac{104}{197}\right)\) | \(e\left(\frac{63}{394}\right)\) | \(e\left(\frac{29}{197}\right)\) | \(e\left(\frac{23}{394}\right)\) | \(e\left(\frac{115}{394}\right)\) | \(e\left(\frac{151}{197}\right)\) | \(e\left(\frac{182}{197}\right)\) | \(e\left(\frac{253}{788}\right)\) |
| \(\chi_{4729}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{394}\right)\) | \(e\left(\frac{169}{394}\right)\) | \(e\left(\frac{89}{197}\right)\) | \(e\left(\frac{321}{394}\right)\) | \(e\left(\frac{129}{197}\right)\) | \(e\left(\frac{211}{394}\right)\) | \(e\left(\frac{267}{394}\right)\) | \(e\left(\frac{169}{197}\right)\) | \(e\left(\frac{8}{197}\right)\) | \(e\left(\frac{351}{788}\right)\) |
| \(\chi_{4729}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{394}\right)\) | \(e\left(\frac{195}{394}\right)\) | \(e\left(\frac{133}{197}\right)\) | \(e\left(\frac{37}{394}\right)\) | \(e\left(\frac{164}{197}\right)\) | \(e\left(\frac{1}{394}\right)\) | \(e\left(\frac{5}{394}\right)\) | \(e\left(\frac{195}{197}\right)\) | \(e\left(\frac{85}{197}\right)\) | \(e\left(\frac{11}{788}\right)\) |
| \(\chi_{4729}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{394}\right)\) | \(e\left(\frac{367}{394}\right)\) | \(e\left(\frac{121}{197}\right)\) | \(e\left(\frac{7}{394}\right)\) | \(e\left(\frac{47}{197}\right)\) | \(e\left(\frac{309}{394}\right)\) | \(e\left(\frac{363}{394}\right)\) | \(e\left(\frac{170}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{247}{788}\right)\) |
| \(\chi_{4729}(243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{394}\right)\) | \(e\left(\frac{353}{394}\right)\) | \(e\left(\frac{67}{197}\right)\) | \(e\left(\frac{69}{394}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{119}{394}\right)\) | \(e\left(\frac{201}{394}\right)\) | \(e\left(\frac{156}{197}\right)\) | \(e\left(\frac{68}{197}\right)\) | \(e\left(\frac{127}{788}\right)\) |
| \(\chi_{4729}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{394}\right)\) | \(e\left(\frac{219}{394}\right)\) | \(e\left(\frac{113}{197}\right)\) | \(e\left(\frac{381}{394}\right)\) | \(e\left(\frac{166}{197}\right)\) | \(e\left(\frac{383}{394}\right)\) | \(e\left(\frac{339}{394}\right)\) | \(e\left(\frac{22}{197}\right)\) | \(e\left(\frac{50}{197}\right)\) | \(e\left(\frac{273}{788}\right)\) |
| \(\chi_{4729}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{319}{394}\right)\) | \(e\left(\frac{287}{394}\right)\) | \(e\left(\frac{122}{197}\right)\) | \(e\left(\frac{305}{394}\right)\) | \(e\left(\frac{106}{197}\right)\) | \(e\left(\frac{349}{394}\right)\) | \(e\left(\frac{169}{394}\right)\) | \(e\left(\frac{90}{197}\right)\) | \(e\left(\frac{115}{197}\right)\) | \(e\left(\frac{293}{788}\right)\) |
| \(\chi_{4729}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{394}\right)\) | \(e\left(\frac{333}{394}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{45}{394}\right)\) | \(e\left(\frac{77}{197}\right)\) | \(e\left(\frac{129}{394}\right)\) | \(e\left(\frac{251}{394}\right)\) | \(e\left(\frac{136}{197}\right)\) | \(e\left(\frac{130}{197}\right)\) | \(e\left(\frac{631}{788}\right)\) |
| \(\chi_{4729}(335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{325}{394}\right)\) | \(e\left(\frac{201}{394}\right)\) | \(e\left(\frac{128}{197}\right)\) | \(e\left(\frac{123}{394}\right)\) | \(e\left(\frac{66}{197}\right)\) | \(e\left(\frac{195}{394}\right)\) | \(e\left(\frac{187}{394}\right)\) | \(e\left(\frac{4}{197}\right)\) | \(e\left(\frac{27}{197}\right)\) | \(e\left(\frac{175}{788}\right)\) |
| \(\chi_{4729}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{394}\right)\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{66}{197}\right)\) | \(e\left(\frac{165}{394}\right)\) | \(e\left(\frac{151}{197}\right)\) | \(e\left(\frac{79}{394}\right)\) | \(e\left(\frac{1}{394}\right)\) | \(e\left(\frac{39}{197}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{81}{788}\right)\) |
| \(\chi_{4729}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{394}\right)\) | \(e\left(\frac{229}{394}\right)\) | \(e\left(\frac{39}{197}\right)\) | \(e\left(\frac{393}{394}\right)\) | \(e\left(\frac{134}{197}\right)\) | \(e\left(\frac{181}{394}\right)\) | \(e\left(\frac{117}{394}\right)\) | \(e\left(\frac{32}{197}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{415}{788}\right)\) |
| \(\chi_{4729}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{394}\right)\) | \(e\left(\frac{19}{394}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{141}{394}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{89}{394}\right)\) | \(e\left(\frac{51}{394}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{79}{197}\right)\) | \(e\left(\frac{585}{788}\right)\) |
| \(\chi_{4729}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{394}\right)\) | \(e\left(\frac{351}{394}\right)\) | \(e\left(\frac{3}{197}\right)\) | \(e\left(\frac{303}{394}\right)\) | \(e\left(\frac{177}{197}\right)\) | \(e\left(\frac{317}{394}\right)\) | \(e\left(\frac{9}{394}\right)\) | \(e\left(\frac{154}{197}\right)\) | \(e\left(\frac{153}{197}\right)\) | \(e\left(\frac{729}{788}\right)\) |
| \(\chi_{4729}(394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{394}\right)\) | \(e\left(\frac{139}{394}\right)\) | \(e\left(\frac{114}{197}\right)\) | \(e\left(\frac{285}{394}\right)\) | \(e\left(\frac{28}{197}\right)\) | \(e\left(\frac{29}{394}\right)\) | \(e\left(\frac{145}{394}\right)\) | \(e\left(\frac{139}{197}\right)\) | \(e\left(\frac{101}{197}\right)\) | \(e\left(\frac{713}{788}\right)\) |