from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(394))
M = H._module
chi = DirichletCharacter(H, M([106]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,4729))
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: | \(197\) | 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: | $\Q(\zeta_{197})$ |
| Fixed field: | Number field defined by a degree 197 polynomial (not computed) |
First 31 of 196 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4729}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{197}\right)\) | \(e\left(\frac{145}{197}\right)\) | \(e\left(\frac{21}{197}\right)\) | \(e\left(\frac{174}{197}\right)\) | \(e\left(\frac{57}{197}\right)\) | \(e\left(\frac{26}{197}\right)\) | \(e\left(\frac{130}{197}\right)\) | \(e\left(\frac{93}{197}\right)\) | \(e\left(\frac{86}{197}\right)\) | \(e\left(\frac{143}{197}\right)\) |
| \(\chi_{4729}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{197}\right)\) | \(e\left(\frac{16}{197}\right)\) | \(e\left(\frac{39}{197}\right)\) | \(e\left(\frac{98}{197}\right)\) | \(e\left(\frac{134}{197}\right)\) | \(e\left(\frac{189}{197}\right)\) | \(e\left(\frac{157}{197}\right)\) | \(e\left(\frac{32}{197}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{153}{197}\right)\) |
| \(\chi_{4729}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{197}\right)\) | \(e\left(\frac{84}{197}\right)\) | \(e\left(\frac{57}{197}\right)\) | \(e\left(\frac{22}{197}\right)\) | \(e\left(\frac{14}{197}\right)\) | \(e\left(\frac{155}{197}\right)\) | \(e\left(\frac{184}{197}\right)\) | \(e\left(\frac{168}{197}\right)\) | \(e\left(\frac{149}{197}\right)\) | \(e\left(\frac{163}{197}\right)\) |
| \(\chi_{4729}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{136}{197}\right)\) | \(e\left(\frac{152}{197}\right)\) | \(e\left(\frac{75}{197}\right)\) | \(e\left(\frac{143}{197}\right)\) | \(e\left(\frac{91}{197}\right)\) | \(e\left(\frac{121}{197}\right)\) | \(e\left(\frac{14}{197}\right)\) | \(e\left(\frac{107}{197}\right)\) | \(e\left(\frac{82}{197}\right)\) | \(e\left(\frac{173}{197}\right)\) |
| \(\chi_{4729}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{197}\right)\) | \(e\left(\frac{23}{197}\right)\) | \(e\left(\frac{93}{197}\right)\) | \(e\left(\frac{67}{197}\right)\) | \(e\left(\frac{168}{197}\right)\) | \(e\left(\frac{87}{197}\right)\) | \(e\left(\frac{41}{197}\right)\) | \(e\left(\frac{46}{197}\right)\) | \(e\left(\frac{15}{197}\right)\) | \(e\left(\frac{183}{197}\right)\) |
| \(\chi_{4729}(250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{197}\right)\) | \(e\left(\frac{185}{197}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{25}{197}\right)\) | \(e\left(\frac{195}{197}\right)\) | \(e\left(\frac{6}{197}\right)\) | \(e\left(\frac{30}{197}\right)\) | \(e\left(\frac{173}{197}\right)\) | \(e\left(\frac{35}{197}\right)\) | \(e\left(\frac{33}{197}\right)\) |
| \(\chi_{4729}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{197}\right)\) | \(e\left(\frac{86}{197}\right)\) | \(e\left(\frac{185}{197}\right)\) | \(e\left(\frac{182}{197}\right)\) | \(e\left(\frac{80}{197}\right)\) | \(e\left(\frac{154}{197}\right)\) | \(e\left(\frac{179}{197}\right)\) | \(e\left(\frac{172}{197}\right)\) | \(e\left(\frac{176}{197}\right)\) | \(e\left(\frac{59}{197}\right)\) |
| \(\chi_{4729}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{197}\right)\) | \(e\left(\frac{106}{197}\right)\) | \(e\left(\frac{86}{197}\right)\) | \(e\left(\frac{9}{197}\right)\) | \(e\left(\frac{149}{197}\right)\) | \(e\left(\frac{144}{197}\right)\) | \(e\left(\frac{129}{197}\right)\) | \(e\left(\frac{15}{197}\right)\) | \(e\left(\frac{52}{197}\right)\) | \(e\left(\frac{4}{197}\right)\) |
| \(\chi_{4729}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{197}\right)\) | \(e\left(\frac{93}{197}\right)\) | \(e\left(\frac{42}{197}\right)\) | \(e\left(\frac{151}{197}\right)\) | \(e\left(\frac{114}{197}\right)\) | \(e\left(\frac{52}{197}\right)\) | \(e\left(\frac{63}{197}\right)\) | \(e\left(\frac{186}{197}\right)\) | \(e\left(\frac{172}{197}\right)\) | \(e\left(\frac{89}{197}\right)\) |
| \(\chi_{4729}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{197}\right)\) | \(e\left(\frac{162}{197}\right)\) | \(e\left(\frac{124}{197}\right)\) | \(e\left(\frac{155}{197}\right)\) | \(e\left(\frac{27}{197}\right)\) | \(e\left(\frac{116}{197}\right)\) | \(e\left(\frac{186}{197}\right)\) | \(e\left(\frac{127}{197}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{47}{197}\right)\) |
| \(\chi_{4729}(314,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{150}{197}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{103}{197}\right)\) | \(e\left(\frac{178}{197}\right)\) | \(e\left(\frac{167}{197}\right)\) | \(e\left(\frac{90}{197}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{34}{197}\right)\) | \(e\left(\frac{131}{197}\right)\) | \(e\left(\frac{101}{197}\right)\) |
| \(\chi_{4729}(319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{197}\right)\) | \(e\left(\frac{108}{197}\right)\) | \(e\left(\frac{17}{197}\right)\) | \(e\left(\frac{169}{197}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{143}{197}\right)\) | \(e\left(\frac{124}{197}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{79}{197}\right)\) | \(e\left(\frac{97}{197}\right)\) |
| \(\chi_{4729}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{197}\right)\) | \(e\left(\frac{137}{197}\right)\) | \(e\left(\frac{100}{197}\right)\) | \(e\left(\frac{125}{197}\right)\) | \(e\left(\frac{187}{197}\right)\) | \(e\left(\frac{30}{197}\right)\) | \(e\left(\frac{150}{197}\right)\) | \(e\left(\frac{77}{197}\right)\) | \(e\left(\frac{175}{197}\right)\) | \(e\left(\frac{165}{197}\right)\) |
| \(\chi_{4729}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{38}{197}\right)\) | \(e\left(\frac{146}{197}\right)\) | \(e\left(\frac{75}{197}\right)\) | \(e\left(\frac{169}{197}\right)\) | \(e\left(\frac{57}{197}\right)\) | \(e\left(\frac{112}{197}\right)\) | \(e\left(\frac{165}{197}\right)\) | \(e\left(\frac{43}{197}\right)\) |
| \(\chi_{4729}(384,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{197}\right)\) | \(e\left(\frac{161}{197}\right)\) | \(e\left(\frac{60}{197}\right)\) | \(e\left(\frac{75}{197}\right)\) | \(e\left(\frac{191}{197}\right)\) | \(e\left(\frac{18}{197}\right)\) | \(e\left(\frac{90}{197}\right)\) | \(e\left(\frac{125}{197}\right)\) | \(e\left(\frac{105}{197}\right)\) | \(e\left(\frac{99}{197}\right)\) |
| \(\chi_{4729}(471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{197}\right)\) | \(e\left(\frac{85}{197}\right)\) | \(e\left(\frac{121}{197}\right)\) | \(e\left(\frac{102}{197}\right)\) | \(e\left(\frac{47}{197}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{83}{197}\right)\) | \(e\left(\frac{170}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{111}{197}\right)\) |
| \(\chi_{4729}(490,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{197}\right)\) | \(e\left(\frac{195}{197}\right)\) | \(e\left(\frac{69}{197}\right)\) | \(e\left(\frac{37}{197}\right)\) | \(e\left(\frac{131}{197}\right)\) | \(e\left(\frac{1}{197}\right)\) | \(e\left(\frac{5}{197}\right)\) | \(e\left(\frac{193}{197}\right)\) | \(e\left(\frac{170}{197}\right)\) | \(e\left(\frac{104}{197}\right)\) |
| \(\chi_{4729}(497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{197}\right)\) | \(e\left(\frac{180}{197}\right)\) | \(e\left(\frac{94}{197}\right)\) | \(e\left(\frac{19}{197}\right)\) | \(e\left(\frac{30}{197}\right)\) | \(e\left(\frac{107}{197}\right)\) | \(e\left(\frac{141}{197}\right)\) | \(e\left(\frac{163}{197}\right)\) | \(e\left(\frac{66}{197}\right)\) | \(e\left(\frac{96}{197}\right)\) |
| \(\chi_{4729}(576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{197}\right)\) | \(e\left(\frac{32}{197}\right)\) | \(e\left(\frac{78}{197}\right)\) | \(e\left(\frac{196}{197}\right)\) | \(e\left(\frac{71}{197}\right)\) | \(e\left(\frac{181}{197}\right)\) | \(e\left(\frac{117}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{38}{197}\right)\) | \(e\left(\frac{109}{197}\right)\) |
| \(\chi_{4729}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{197}\right)\) | \(e\left(\frac{147}{197}\right)\) | \(e\left(\frac{149}{197}\right)\) | \(e\left(\frac{137}{197}\right)\) | \(e\left(\frac{123}{197}\right)\) | \(e\left(\frac{25}{197}\right)\) | \(e\left(\frac{125}{197}\right)\) | \(e\left(\frac{97}{197}\right)\) | \(e\left(\frac{113}{197}\right)\) | \(e\left(\frac{39}{197}\right)\) |
| \(\chi_{4729}(665,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{132}{197}\right)\) | \(e\left(\frac{78}{197}\right)\) | \(e\left(\frac{67}{197}\right)\) | \(e\left(\frac{133}{197}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{158}{197}\right)\) | \(e\left(\frac{2}{197}\right)\) | \(e\left(\frac{156}{197}\right)\) | \(e\left(\frac{68}{197}\right)\) | \(e\left(\frac{81}{197}\right)\) |
| \(\chi_{4729}(682,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{197}\right)\) | \(e\left(\frac{105}{197}\right)\) | \(e\left(\frac{22}{197}\right)\) | \(e\left(\frac{126}{197}\right)\) | \(e\left(\frac{116}{197}\right)\) | \(e\left(\frac{46}{197}\right)\) | \(e\left(\frac{33}{197}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{137}{197}\right)\) | \(e\left(\frac{56}{197}\right)\) |
| \(\chi_{4729}(734,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{197}\right)\) | \(e\left(\frac{95}{197}\right)\) | \(e\left(\frac{170}{197}\right)\) | \(e\left(\frac{114}{197}\right)\) | \(e\left(\frac{180}{197}\right)\) | \(e\left(\frac{51}{197}\right)\) | \(e\left(\frac{58}{197}\right)\) | \(e\left(\frac{190}{197}\right)\) | \(e\left(\frac{2}{197}\right)\) | \(e\left(\frac{182}{197}\right)\) |
| \(\chi_{4729}(735,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{142}{197}\right)\) | \(e\left(\frac{66}{197}\right)\) | \(e\left(\frac{87}{197}\right)\) | \(e\left(\frac{158}{197}\right)\) | \(e\left(\frac{11}{197}\right)\) | \(e\left(\frac{164}{197}\right)\) | \(e\left(\frac{32}{197}\right)\) | \(e\left(\frac{132}{197}\right)\) | \(e\left(\frac{103}{197}\right)\) | \(e\left(\frac{114}{197}\right)\) |
| \(\chi_{4729}(739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{197}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{194}{197}\right)\) | \(e\left(\frac{144}{197}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{137}{197}\right)\) | \(e\left(\frac{94}{197}\right)\) | \(e\left(\frac{43}{197}\right)\) | \(e\left(\frac{44}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) |
| \(\chi_{4729}(746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{156}{197}\right)\) | \(e\left(\frac{195}{197}\right)\) | \(e\left(\frac{142}{197}\right)\) | \(e\left(\frac{165}{197}\right)\) | \(e\left(\frac{37}{197}\right)\) | \(e\left(\frac{128}{197}\right)\) | \(e\left(\frac{76}{197}\right)\) | \(e\left(\frac{21}{197}\right)\) |
| \(\chi_{4729}(788,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{197}\right)\) | \(e\left(\frac{155}{197}\right)\) | \(e\left(\frac{70}{197}\right)\) | \(e\left(\frac{186}{197}\right)\) | \(e\left(\frac{190}{197}\right)\) | \(e\left(\frac{21}{197}\right)\) | \(e\left(\frac{105}{197}\right)\) | \(e\left(\frac{113}{197}\right)\) | \(e\left(\frac{24}{197}\right)\) | \(e\left(\frac{17}{197}\right)\) |
| \(\chi_{4729}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{197}\right)\) | \(e\left(\frac{139}{197}\right)\) | \(e\left(\frac{31}{197}\right)\) | \(e\left(\frac{88}{197}\right)\) | \(e\left(\frac{56}{197}\right)\) | \(e\left(\frac{29}{197}\right)\) | \(e\left(\frac{145}{197}\right)\) | \(e\left(\frac{81}{197}\right)\) | \(e\left(\frac{5}{197}\right)\) | \(e\left(\frac{61}{197}\right)\) |
| \(\chi_{4729}(857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{197}\right)\) | \(e\left(\frac{125}{197}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{150}{197}\right)\) | \(e\left(\frac{185}{197}\right)\) | \(e\left(\frac{36}{197}\right)\) | \(e\left(\frac{180}{197}\right)\) | \(e\left(\frac{53}{197}\right)\) | \(e\left(\frac{13}{197}\right)\) | \(e\left(\frac{1}{197}\right)\) |
| \(\chi_{4729}(864,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{197}\right)\) | \(e\left(\frac{100}{197}\right)\) | \(e\left(\frac{96}{197}\right)\) | \(e\left(\frac{120}{197}\right)\) | \(e\left(\frac{148}{197}\right)\) | \(e\left(\frac{147}{197}\right)\) | \(e\left(\frac{144}{197}\right)\) | \(e\left(\frac{3}{197}\right)\) | \(e\left(\frac{168}{197}\right)\) | \(e\left(\frac{119}{197}\right)\) |
| \(\chi_{4729}(907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{138}{197}\right)\) | \(e\left(\frac{189}{197}\right)\) | \(e\left(\frac{79}{197}\right)\) | \(e\left(\frac{148}{197}\right)\) | \(e\left(\frac{130}{197}\right)\) | \(e\left(\frac{4}{197}\right)\) | \(e\left(\frac{20}{197}\right)\) | \(e\left(\frac{181}{197}\right)\) | \(e\left(\frac{89}{197}\right)\) | \(e\left(\frac{22}{197}\right)\) |