from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1077, base_ring=CyclotomicField(358))
M = H._module
chi = DirichletCharacter(H, M([179,315]))
chi.galois_orbit()
[g,chi] = znchar(Mod(14,1077))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1077\) | |
| Conductor: | \(1077\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(358\) | 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: | $\Q(\zeta_{179})$ |
| Fixed field: | Number field defined by a degree 358 polynomial (not computed) |
First 31 of 178 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1077}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{358}\right)\) | \(e\left(\frac{102}{179}\right)\) | \(e\left(\frac{117}{358}\right)\) | \(e\left(\frac{315}{358}\right)\) | \(e\left(\frac{127}{358}\right)\) | \(e\left(\frac{20}{179}\right)\) | \(e\left(\frac{163}{358}\right)\) | \(e\left(\frac{135}{358}\right)\) | \(e\left(\frac{119}{179}\right)\) | \(e\left(\frac{25}{179}\right)\) |
| \(\chi_{1077}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{255}{358}\right)\) | \(e\left(\frac{76}{179}\right)\) | \(e\left(\frac{203}{358}\right)\) | \(e\left(\frac{161}{358}\right)\) | \(e\left(\frac{49}{358}\right)\) | \(e\left(\frac{50}{179}\right)\) | \(e\left(\frac{139}{358}\right)\) | \(e\left(\frac{69}{358}\right)\) | \(e\left(\frac{29}{179}\right)\) | \(e\left(\frac{152}{179}\right)\) |
| \(\chi_{1077}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{358}\right)\) | \(e\left(\frac{137}{179}\right)\) | \(e\left(\frac{15}{358}\right)\) | \(e\left(\frac{123}{358}\right)\) | \(e\left(\frac{53}{358}\right)\) | \(e\left(\frac{76}{179}\right)\) | \(e\left(\frac{333}{358}\right)\) | \(e\left(\frac{155}{358}\right)\) | \(e\left(\frac{130}{179}\right)\) | \(e\left(\frac{95}{179}\right)\) |
| \(\chi_{1077}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{358}\right)\) | \(e\left(\frac{134}{179}\right)\) | \(e\left(\frac{259}{358}\right)\) | \(e\left(\frac{119}{358}\right)\) | \(e\left(\frac{223}{358}\right)\) | \(e\left(\frac{107}{179}\right)\) | \(e\left(\frac{165}{358}\right)\) | \(e\left(\frac{51}{358}\right)\) | \(e\left(\frac{37}{179}\right)\) | \(e\left(\frac{89}{179}\right)\) |
| \(\chi_{1077}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{358}\right)\) | \(e\left(\frac{40}{179}\right)\) | \(e\left(\frac{267}{358}\right)\) | \(e\left(\frac{113}{358}\right)\) | \(e\left(\frac{299}{358}\right)\) | \(e\left(\frac{64}{179}\right)\) | \(e\left(\frac{271}{358}\right)\) | \(e\left(\frac{253}{358}\right)\) | \(e\left(\frac{166}{179}\right)\) | \(e\left(\frac{80}{179}\right)\) |
| \(\chi_{1077}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{358}\right)\) | \(e\left(\frac{28}{179}\right)\) | \(e\left(\frac{169}{358}\right)\) | \(e\left(\frac{97}{358}\right)\) | \(e\left(\frac{263}{358}\right)\) | \(e\left(\frac{9}{179}\right)\) | \(e\left(\frac{315}{358}\right)\) | \(e\left(\frac{195}{358}\right)\) | \(e\left(\frac{152}{179}\right)\) | \(e\left(\frac{56}{179}\right)\) |
| \(\chi_{1077}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{358}\right)\) | \(e\left(\frac{36}{179}\right)\) | \(e\left(\frac{115}{358}\right)\) | \(e\left(\frac{227}{358}\right)\) | \(e\left(\frac{287}{358}\right)\) | \(e\left(\frac{165}{179}\right)\) | \(e\left(\frac{47}{358}\right)\) | \(e\left(\frac{353}{358}\right)\) | \(e\left(\frac{42}{179}\right)\) | \(e\left(\frac{72}{179}\right)\) |
| \(\chi_{1077}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{358}\right)\) | \(e\left(\frac{161}{179}\right)\) | \(e\left(\frac{211}{358}\right)\) | \(e\left(\frac{155}{358}\right)\) | \(e\left(\frac{125}{358}\right)\) | \(e\left(\frac{7}{179}\right)\) | \(e\left(\frac{245}{358}\right)\) | \(e\left(\frac{271}{358}\right)\) | \(e\left(\frac{158}{179}\right)\) | \(e\left(\frac{143}{179}\right)\) |
| \(\chi_{1077}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{358}\right)\) | \(e\left(\frac{92}{179}\right)\) | \(e\left(\frac{95}{358}\right)\) | \(e\left(\frac{63}{358}\right)\) | \(e\left(\frac{97}{358}\right)\) | \(e\left(\frac{4}{179}\right)\) | \(e\left(\frac{319}{358}\right)\) | \(e\left(\frac{27}{358}\right)\) | \(e\left(\frac{167}{179}\right)\) | \(e\left(\frac{5}{179}\right)\) |
| \(\chi_{1077}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{358}\right)\) | \(e\left(\frac{108}{179}\right)\) | \(e\left(\frac{345}{358}\right)\) | \(e\left(\frac{323}{358}\right)\) | \(e\left(\frac{145}{358}\right)\) | \(e\left(\frac{137}{179}\right)\) | \(e\left(\frac{141}{358}\right)\) | \(e\left(\frac{343}{358}\right)\) | \(e\left(\frac{126}{179}\right)\) | \(e\left(\frac{37}{179}\right)\) |
| \(\chi_{1077}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{358}\right)\) | \(e\left(\frac{114}{179}\right)\) | \(e\left(\frac{215}{358}\right)\) | \(e\left(\frac{331}{358}\right)\) | \(e\left(\frac{163}{358}\right)\) | \(e\left(\frac{75}{179}\right)\) | \(e\left(\frac{119}{358}\right)\) | \(e\left(\frac{193}{358}\right)\) | \(e\left(\frac{133}{179}\right)\) | \(e\left(\frac{49}{179}\right)\) |
| \(\chi_{1077}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{358}\right)\) | \(e\left(\frac{77}{179}\right)\) | \(e\left(\frac{241}{358}\right)\) | \(e\left(\frac{43}{358}\right)\) | \(e\left(\frac{231}{358}\right)\) | \(e\left(\frac{159}{179}\right)\) | \(e\left(\frac{195}{358}\right)\) | \(e\left(\frac{223}{358}\right)\) | \(e\left(\frac{60}{179}\right)\) | \(e\left(\frac{154}{179}\right)\) |
| \(\chi_{1077}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{358}\right)\) | \(e\left(\frac{37}{179}\right)\) | \(e\left(\frac{153}{358}\right)\) | \(e\left(\frac{109}{358}\right)\) | \(e\left(\frac{111}{358}\right)\) | \(e\left(\frac{95}{179}\right)\) | \(e\left(\frac{103}{358}\right)\) | \(e\left(\frac{149}{358}\right)\) | \(e\left(\frac{73}{179}\right)\) | \(e\left(\frac{74}{179}\right)\) |
| \(\chi_{1077}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{358}\right)\) | \(e\left(\frac{115}{179}\right)\) | \(e\left(\frac{253}{358}\right)\) | \(e\left(\frac{213}{358}\right)\) | \(e\left(\frac{345}{358}\right)\) | \(e\left(\frac{5}{179}\right)\) | \(e\left(\frac{175}{358}\right)\) | \(e\left(\frac{347}{358}\right)\) | \(e\left(\frac{164}{179}\right)\) | \(e\left(\frac{51}{179}\right)\) |
| \(\chi_{1077}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{358}\right)\) | \(e\left(\frac{116}{179}\right)\) | \(e\left(\frac{291}{358}\right)\) | \(e\left(\frac{95}{358}\right)\) | \(e\left(\frac{169}{358}\right)\) | \(e\left(\frac{114}{179}\right)\) | \(e\left(\frac{231}{358}\right)\) | \(e\left(\frac{143}{358}\right)\) | \(e\left(\frac{16}{179}\right)\) | \(e\left(\frac{53}{179}\right)\) |
| \(\chi_{1077}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{358}\right)\) | \(e\left(\frac{72}{179}\right)\) | \(e\left(\frac{51}{358}\right)\) | \(e\left(\frac{275}{358}\right)\) | \(e\left(\frac{37}{358}\right)\) | \(e\left(\frac{151}{179}\right)\) | \(e\left(\frac{273}{358}\right)\) | \(e\left(\frac{169}{358}\right)\) | \(e\left(\frac{84}{179}\right)\) | \(e\left(\frac{144}{179}\right)\) |
| \(\chi_{1077}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{358}\right)\) | \(e\left(\frac{10}{179}\right)\) | \(e\left(\frac{201}{358}\right)\) | \(e\left(\frac{73}{358}\right)\) | \(e\left(\frac{209}{358}\right)\) | \(e\left(\frac{16}{179}\right)\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{287}{358}\right)\) | \(e\left(\frac{131}{179}\right)\) | \(e\left(\frac{20}{179}\right)\) |
| \(\chi_{1077}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{358}\right)\) | \(e\left(\frac{62}{179}\right)\) | \(e\left(\frac{29}{358}\right)\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{7}{358}\right)\) | \(e\left(\frac{135}{179}\right)\) | \(e\left(\frac{71}{358}\right)\) | \(e\left(\frac{61}{358}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{124}{179}\right)\) |
| \(\chi_{1077}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{358}\right)\) | \(e\left(\frac{71}{179}\right)\) | \(e\left(\frac{13}{358}\right)\) | \(e\left(\frac{35}{358}\right)\) | \(e\left(\frac{213}{358}\right)\) | \(e\left(\frac{42}{179}\right)\) | \(e\left(\frac{217}{358}\right)\) | \(e\left(\frac{15}{358}\right)\) | \(e\left(\frac{53}{179}\right)\) | \(e\left(\frac{142}{179}\right)\) |
| \(\chi_{1077}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{358}\right)\) | \(e\left(\frac{96}{179}\right)\) | \(e\left(\frac{247}{358}\right)\) | \(e\left(\frac{307}{358}\right)\) | \(e\left(\frac{109}{358}\right)\) | \(e\left(\frac{82}{179}\right)\) | \(e\left(\frac{185}{358}\right)\) | \(e\left(\frac{285}{358}\right)\) | \(e\left(\frac{112}{179}\right)\) | \(e\left(\frac{13}{179}\right)\) |
| \(\chi_{1077}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{358}\right)\) | \(e\left(\frac{85}{179}\right)\) | \(e\left(\frac{187}{358}\right)\) | \(e\left(\frac{173}{358}\right)\) | \(e\left(\frac{255}{358}\right)\) | \(e\left(\frac{136}{179}\right)\) | \(e\left(\frac{285}{358}\right)\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{129}{179}\right)\) | \(e\left(\frac{170}{179}\right)\) |
| \(\chi_{1077}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{358}\right)\) | \(e\left(\frac{98}{179}\right)\) | \(e\left(\frac{323}{358}\right)\) | \(e\left(\frac{71}{358}\right)\) | \(e\left(\frac{115}{358}\right)\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{297}{358}\right)\) | \(e\left(\frac{235}{358}\right)\) | \(e\left(\frac{174}{179}\right)\) | \(e\left(\frac{17}{179}\right)\) |
| \(\chi_{1077}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{351}{358}\right)\) | \(e\left(\frac{172}{179}\right)\) | \(e\left(\frac{271}{358}\right)\) | \(e\left(\frac{289}{358}\right)\) | \(e\left(\frac{337}{358}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{145}{358}\right)\) | \(e\left(\frac{175}{358}\right)\) | \(e\left(\frac{141}{179}\right)\) | \(e\left(\frac{165}{179}\right)\) |
| \(\chi_{1077}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{358}\right)\) | \(e\left(\frac{68}{179}\right)\) | \(e\left(\frac{257}{358}\right)\) | \(e\left(\frac{31}{358}\right)\) | \(e\left(\frac{25}{358}\right)\) | \(e\left(\frac{73}{179}\right)\) | \(e\left(\frac{49}{358}\right)\) | \(e\left(\frac{269}{358}\right)\) | \(e\left(\frac{139}{179}\right)\) | \(e\left(\frac{136}{179}\right)\) |
| \(\chi_{1077}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{358}\right)\) | \(e\left(\frac{51}{179}\right)\) | \(e\left(\frac{327}{358}\right)\) | \(e\left(\frac{247}{358}\right)\) | \(e\left(\frac{153}{358}\right)\) | \(e\left(\frac{10}{179}\right)\) | \(e\left(\frac{171}{358}\right)\) | \(e\left(\frac{157}{358}\right)\) | \(e\left(\frac{149}{179}\right)\) | \(e\left(\frac{102}{179}\right)\) |
| \(\chi_{1077}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{358}\right)\) | \(e\left(\frac{153}{179}\right)\) | \(e\left(\frac{265}{358}\right)\) | \(e\left(\frac{25}{358}\right)\) | \(e\left(\frac{101}{358}\right)\) | \(e\left(\frac{30}{179}\right)\) | \(e\left(\frac{155}{358}\right)\) | \(e\left(\frac{113}{358}\right)\) | \(e\left(\frac{89}{179}\right)\) | \(e\left(\frac{127}{179}\right)\) |
| \(\chi_{1077}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{303}{358}\right)\) | \(e\left(\frac{124}{179}\right)\) | \(e\left(\frac{237}{358}\right)\) | \(e\left(\frac{225}{358}\right)\) | \(e\left(\frac{193}{358}\right)\) | \(e\left(\frac{91}{179}\right)\) | \(e\left(\frac{321}{358}\right)\) | \(e\left(\frac{301}{358}\right)\) | \(e\left(\frac{85}{179}\right)\) | \(e\left(\frac{69}{179}\right)\) |
| \(\chi_{1077}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{358}\right)\) | \(e\left(\frac{9}{179}\right)\) | \(e\left(\frac{163}{358}\right)\) | \(e\left(\frac{191}{358}\right)\) | \(e\left(\frac{27}{358}\right)\) | \(e\left(\frac{86}{179}\right)\) | \(e\left(\frac{325}{358}\right)\) | \(e\left(\frac{133}{358}\right)\) | \(e\left(\frac{100}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) |
| \(\chi_{1077}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{358}\right)\) | \(e\left(\frac{31}{179}\right)\) | \(e\left(\frac{283}{358}\right)\) | \(e\left(\frac{101}{358}\right)\) | \(e\left(\frac{93}{358}\right)\) | \(e\left(\frac{157}{179}\right)\) | \(e\left(\frac{125}{358}\right)\) | \(e\left(\frac{299}{358}\right)\) | \(e\left(\frac{66}{179}\right)\) | \(e\left(\frac{62}{179}\right)\) |
| \(\chi_{1077}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{358}\right)\) | \(e\left(\frac{33}{179}\right)\) | \(e\left(\frac{1}{358}\right)\) | \(e\left(\frac{223}{358}\right)\) | \(e\left(\frac{99}{358}\right)\) | \(e\left(\frac{17}{179}\right)\) | \(e\left(\frac{237}{358}\right)\) | \(e\left(\frac{249}{358}\right)\) | \(e\left(\frac{128}{179}\right)\) | \(e\left(\frac{66}{179}\right)\) |
| \(\chi_{1077}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{358}\right)\) | \(e\left(\frac{170}{179}\right)\) | \(e\left(\frac{195}{358}\right)\) | \(e\left(\frac{167}{358}\right)\) | \(e\left(\frac{331}{358}\right)\) | \(e\left(\frac{93}{179}\right)\) | \(e\left(\frac{33}{358}\right)\) | \(e\left(\frac{225}{358}\right)\) | \(e\left(\frac{79}{179}\right)\) | \(e\left(\frac{161}{179}\right)\) |