from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,967))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(967\) | |
| Conductor: | \(967\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(966\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{483})$ |
| Fixed field: | Number field defined by a degree 966 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{967}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{483}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{374}{483}\right)\) | \(e\left(\frac{1}{966}\right)\) | \(e\left(\frac{611}{966}\right)\) | \(e\left(\frac{67}{966}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{148}{161}\right)\) |
| \(\chi_{967}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{483}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{55}{483}\right)\) | \(e\left(\frac{611}{966}\right)\) | \(e\left(\frac{445}{966}\right)\) | \(e\left(\frac{365}{966}\right)\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) |
| \(\chi_{967}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{454}{483}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{425}{483}\right)\) | \(e\left(\frac{67}{966}\right)\) | \(e\left(\frac{365}{966}\right)\) | \(e\left(\frac{625}{966}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{95}{161}\right)\) |
| \(\chi_{967}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{172}{483}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{344}{483}\right)\) | \(e\left(\frac{19}{966}\right)\) | \(e\left(\frac{17}{966}\right)\) | \(e\left(\frac{307}{966}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{75}{161}\right)\) |
| \(\chi_{967}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{418}{483}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{353}{483}\right)\) | \(e\left(\frac{883}{966}\right)\) | \(e\left(\frac{485}{966}\right)\) | \(e\left(\frac{235}{966}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) |
| \(\chi_{967}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{483}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{226}{483}\right)\) | \(e\left(\frac{605}{966}\right)\) | \(e\left(\frac{643}{966}\right)\) | \(e\left(\frac{929}{966}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{24}{161}\right)\) |
| \(\chi_{967}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{260}{483}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{37}{483}\right)\) | \(e\left(\frac{815}{966}\right)\) | \(e\left(\frac{475}{966}\right)\) | \(e\left(\frac{509}{966}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{31}{161}\right)\) |
| \(\chi_{967}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{483}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{215}{483}\right)\) | \(e\left(\frac{193}{966}\right)\) | \(e\left(\frac{71}{966}\right)\) | \(e\left(\frac{373}{966}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{67}{161}\right)\) |
| \(\chi_{967}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{483}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{32}{483}\right)\) | \(e\left(\frac{13}{966}\right)\) | \(e\left(\frac{215}{966}\right)\) | \(e\left(\frac{871}{966}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{61}{161}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{153}{161}\right)\) |
| \(\chi_{967}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{483}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{275}{483}\right)\) | \(e\left(\frac{157}{966}\right)\) | \(e\left(\frac{293}{966}\right)\) | \(e\left(\frac{859}{966}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{52}{161}\right)\) |
| \(\chi_{967}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{483}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{94}{483}\right)\) | \(e\left(\frac{491}{966}\right)\) | \(e\left(\frac{541}{966}\right)\) | \(e\left(\frac{53}{966}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{57}{161}\right)\) |
| \(\chi_{967}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{436}{483}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{389}{483}\right)\) | \(e\left(\frac{475}{966}\right)\) | \(e\left(\frac{425}{966}\right)\) | \(e\left(\frac{913}{966}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{104}{161}\right)\) |
| \(\chi_{967}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{146}{483}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{292}{483}\right)\) | \(e\left(\frac{179}{966}\right)\) | \(e\left(\frac{211}{966}\right)\) | \(e\left(\frac{401}{966}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) |
| \(\chi_{967}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{483}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{278}{483}\right)\) | \(e\left(\frac{445}{966}\right)\) | \(e\left(\frac{449}{966}\right)\) | \(e\left(\frac{835}{966}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{11}{161}\right)\) |
| \(\chi_{967}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{483}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{439}{483}\right)\) | \(e\left(\frac{767}{966}\right)\) | \(e\left(\frac{127}{966}\right)\) | \(e\left(\frac{191}{966}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{11}{161}\right)\) |
| \(\chi_{967}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{483}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{326}{483}\right)\) | \(e\left(\frac{223}{966}\right)\) | \(e\left(\frac{47}{966}\right)\) | \(e\left(\frac{451}{966}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) |
| \(\chi_{967}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{206}{483}\right)\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{412}{483}\right)\) | \(e\left(\frac{107}{966}\right)\) | \(e\left(\frac{655}{966}\right)\) | \(e\left(\frac{407}{966}\right)\) | \(e\left(\frac{45}{161}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{58}{161}\right)\) |
| \(\chi_{967}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{220}{483}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{440}{483}\right)\) | \(e\left(\frac{541}{966}\right)\) | \(e\left(\frac{179}{966}\right)\) | \(e\left(\frac{505}{966}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{51}{161}\right)\) |
| \(\chi_{967}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{483}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{346}{483}\right)\) | \(e\left(\frac{533}{966}\right)\) | \(e\left(\frac{121}{966}\right)\) | \(e\left(\frac{935}{966}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{86}{161}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{155}{161}\right)\) |
| \(\chi_{967}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{483}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{31}{483}\right)\) | \(e\left(\frac{239}{966}\right)\) | \(e\left(\frac{163}{966}\right)\) | \(e\left(\frac{557}{966}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) |
| \(\chi_{967}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{358}{483}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{233}{483}\right)\) | \(e\left(\frac{955}{966}\right)\) | \(e\left(\frac{41}{966}\right)\) | \(e\left(\frac{229}{966}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{97}{161}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{143}{161}\right)\) |
| \(\chi_{967}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{424}{483}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{365}{483}\right)\) | \(e\left(\frac{103}{966}\right)\) | \(e\left(\frac{143}{966}\right)\) | \(e\left(\frac{139}{966}\right)\) | \(e\left(\frac{102}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{110}{161}\right)\) |
| \(\chi_{967}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{218}{483}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{436}{483}\right)\) | \(e\left(\frac{479}{966}\right)\) | \(e\left(\frac{937}{966}\right)\) | \(e\left(\frac{215}{966}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{52}{161}\right)\) |
| \(\chi_{967}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{483}\right)\) | \(e\left(\frac{137}{322}\right)\) | \(e\left(\frac{188}{483}\right)\) | \(e\left(\frac{499}{966}\right)\) | \(e\left(\frac{599}{966}\right)\) | \(e\left(\frac{589}{966}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) |
| \(\chi_{967}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{433}{483}\right)\) | \(e\left(\frac{71}{322}\right)\) | \(e\left(\frac{383}{483}\right)\) | \(e\left(\frac{865}{966}\right)\) | \(e\left(\frac{113}{966}\right)\) | \(e\left(\frac{961}{966}\right)\) | \(e\left(\frac{111}{161}\right)\) | \(e\left(\frac{71}{161}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{25}{161}\right)\) |
| \(\chi_{967}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{268}{483}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{53}{483}\right)\) | \(e\left(\frac{97}{966}\right)\) | \(e\left(\frac{341}{966}\right)\) | \(e\left(\frac{703}{966}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) |
| \(\chi_{967}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{176}{483}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{352}{483}\right)\) | \(e\left(\frac{143}{966}\right)\) | \(e\left(\frac{433}{966}\right)\) | \(e\left(\frac{887}{966}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) |
| \(\chi_{967}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{483}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{254}{483}\right)\) | \(e\left(\frac{73}{966}\right)\) | \(e\left(\frac{167}{966}\right)\) | \(e\left(\frac{61}{966}\right)\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) |
| \(\chi_{967}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{483}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{82}{483}\right)\) | \(e\left(\frac{305}{966}\right)\) | \(e\left(\frac{883}{966}\right)\) | \(e\left(\frac{149}{966}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{60}{161}\right)\) |
| \(\chi_{967}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{483}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{80}{483}\right)\) | \(e\left(\frac{757}{966}\right)\) | \(e\left(\frac{779}{966}\right)\) | \(e\left(\frac{487}{966}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{141}{161}\right)\) |
| \(\chi_{967}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{483}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{359}{483}\right)\) | \(e\left(\frac{493}{966}\right)\) | \(e\left(\frac{797}{966}\right)\) | \(e\left(\frac{187}{966}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{31}{161}\right)\) |