from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1259, base_ring=CyclotomicField(1258))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1259))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1259\) | |
Conductor: | \(1259\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(1258\) | 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: | $\Q(\zeta_{629})$ |
Fixed field: | Number field defined by a degree 1258 polynomial (not computed) |
First 31 of 576 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1259}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1258}\right)\) | \(e\left(\frac{548}{629}\right)\) | \(e\left(\frac{1}{629}\right)\) | \(e\left(\frac{38}{629}\right)\) | \(e\left(\frac{1097}{1258}\right)\) | \(e\left(\frac{123}{629}\right)\) | \(e\left(\frac{3}{1258}\right)\) | \(e\left(\frac{467}{629}\right)\) | \(e\left(\frac{77}{1258}\right)\) | \(e\left(\frac{1151}{1258}\right)\) |
\(\chi_{1259}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1097}{1258}\right)\) | \(e\left(\frac{461}{629}\right)\) | \(e\left(\frac{468}{629}\right)\) | \(e\left(\frac{172}{629}\right)\) | \(e\left(\frac{761}{1258}\right)\) | \(e\left(\frac{325}{629}\right)\) | \(e\left(\frac{775}{1258}\right)\) | \(e\left(\frac{293}{629}\right)\) | \(e\left(\frac{183}{1258}\right)\) | \(e\left(\frac{873}{1258}\right)\) |
\(\chi_{1259}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{1258}\right)\) | \(e\left(\frac{386}{629}\right)\) | \(e\left(\frac{3}{629}\right)\) | \(e\left(\frac{114}{629}\right)\) | \(e\left(\frac{775}{1258}\right)\) | \(e\left(\frac{369}{629}\right)\) | \(e\left(\frac{9}{1258}\right)\) | \(e\left(\frac{143}{629}\right)\) | \(e\left(\frac{231}{1258}\right)\) | \(e\left(\frac{937}{1258}\right)\) |
\(\chi_{1259}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{1258}\right)\) | \(e\left(\frac{53}{629}\right)\) | \(e\left(\frac{77}{629}\right)\) | \(e\left(\frac{410}{629}\right)\) | \(e\left(\frac{183}{1258}\right)\) | \(e\left(\frac{36}{629}\right)\) | \(e\left(\frac{231}{1258}\right)\) | \(e\left(\frac{106}{629}\right)\) | \(e\left(\frac{897}{1258}\right)\) | \(e\left(\frac{567}{1258}\right)\) |
\(\chi_{1259}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1151}{1258}\right)\) | \(e\left(\frac{490}{629}\right)\) | \(e\left(\frac{522}{629}\right)\) | \(e\left(\frac{337}{629}\right)\) | \(e\left(\frac{873}{1258}\right)\) | \(e\left(\frac{48}{629}\right)\) | \(e\left(\frac{937}{1258}\right)\) | \(e\left(\frac{351}{629}\right)\) | \(e\left(\frac{567}{1258}\right)\) | \(e\left(\frac{127}{1258}\right)\) |
\(\chi_{1259}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{679}{1258}\right)\) | \(e\left(\frac{353}{629}\right)\) | \(e\left(\frac{50}{629}\right)\) | \(e\left(\frac{13}{629}\right)\) | \(e\left(\frac{127}{1258}\right)\) | \(e\left(\frac{489}{629}\right)\) | \(e\left(\frac{779}{1258}\right)\) | \(e\left(\frac{77}{629}\right)\) | \(e\left(\frac{705}{1258}\right)\) | \(e\left(\frac{311}{1258}\right)\) |
\(\chi_{1259}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{1258}\right)\) | \(e\left(\frac{121}{629}\right)\) | \(e\left(\frac{247}{629}\right)\) | \(e\left(\frac{580}{629}\right)\) | \(e\left(\frac{489}{1258}\right)\) | \(e\left(\frac{189}{629}\right)\) | \(e\left(\frac{741}{1258}\right)\) | \(e\left(\frac{242}{629}\right)\) | \(e\left(\frac{149}{1258}\right)\) | \(e\left(\frac{1247}{1258}\right)\) |
\(\chi_{1259}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{973}{1258}\right)\) | \(e\left(\frac{441}{629}\right)\) | \(e\left(\frac{344}{629}\right)\) | \(e\left(\frac{492}{629}\right)\) | \(e\left(\frac{597}{1258}\right)\) | \(e\left(\frac{169}{629}\right)\) | \(e\left(\frac{403}{1258}\right)\) | \(e\left(\frac{253}{629}\right)\) | \(e\left(\frac{699}{1258}\right)\) | \(e\left(\frac{303}{1258}\right)\) |
\(\chi_{1259}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1099}{1258}\right)\) | \(e\left(\frac{299}{629}\right)\) | \(e\left(\frac{470}{629}\right)\) | \(e\left(\frac{248}{629}\right)\) | \(e\left(\frac{439}{1258}\right)\) | \(e\left(\frac{571}{629}\right)\) | \(e\left(\frac{781}{1258}\right)\) | \(e\left(\frac{598}{629}\right)\) | \(e\left(\frac{337}{1258}\right)\) | \(e\left(\frac{659}{1258}\right)\) |
\(\chi_{1259}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{1258}\right)\) | \(e\left(\frac{550}{629}\right)\) | \(e\left(\frac{265}{629}\right)\) | \(e\left(\frac{6}{629}\right)\) | \(e\left(\frac{107}{1258}\right)\) | \(e\left(\frac{516}{629}\right)\) | \(e\left(\frac{795}{1258}\right)\) | \(e\left(\frac{471}{629}\right)\) | \(e\left(\frac{277}{1258}\right)\) | \(e\left(\frac{579}{1258}\right)\) |
\(\chi_{1259}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{1258}\right)\) | \(e\left(\frac{224}{629}\right)\) | \(e\left(\frac{5}{629}\right)\) | \(e\left(\frac{190}{629}\right)\) | \(e\left(\frac{453}{1258}\right)\) | \(e\left(\frac{615}{629}\right)\) | \(e\left(\frac{15}{1258}\right)\) | \(e\left(\frac{448}{629}\right)\) | \(e\left(\frac{385}{1258}\right)\) | \(e\left(\frac{723}{1258}\right)\) |
\(\chi_{1259}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{989}{1258}\right)\) | \(e\left(\frac{403}{629}\right)\) | \(e\left(\frac{360}{629}\right)\) | \(e\left(\frac{471}{629}\right)\) | \(e\left(\frac{537}{1258}\right)\) | \(e\left(\frac{250}{629}\right)\) | \(e\left(\frac{451}{1258}\right)\) | \(e\left(\frac{177}{629}\right)\) | \(e\left(\frac{673}{1258}\right)\) | \(e\left(\frac{1107}{1258}\right)\) |
\(\chi_{1259}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{1258}\right)\) | \(e\left(\frac{43}{629}\right)\) | \(e\left(\frac{15}{629}\right)\) | \(e\left(\frac{570}{629}\right)\) | \(e\left(\frac{101}{1258}\right)\) | \(e\left(\frac{587}{629}\right)\) | \(e\left(\frac{45}{1258}\right)\) | \(e\left(\frac{86}{629}\right)\) | \(e\left(\frac{1155}{1258}\right)\) | \(e\left(\frac{911}{1258}\right)\) |
\(\chi_{1259}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{517}{1258}\right)\) | \(e\left(\frac{266}{629}\right)\) | \(e\left(\frac{517}{629}\right)\) | \(e\left(\frac{147}{629}\right)\) | \(e\left(\frac{1049}{1258}\right)\) | \(e\left(\frac{62}{629}\right)\) | \(e\left(\frac{293}{1258}\right)\) | \(e\left(\frac{532}{629}\right)\) | \(e\left(\frac{811}{1258}\right)\) | \(e\left(\frac{33}{1258}\right)\) |
\(\chi_{1259}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{1258}\right)\) | \(e\left(\frac{520}{629}\right)\) | \(e\left(\frac{79}{629}\right)\) | \(e\left(\frac{486}{629}\right)\) | \(e\left(\frac{1119}{1258}\right)\) | \(e\left(\frac{282}{629}\right)\) | \(e\left(\frac{237}{1258}\right)\) | \(e\left(\frac{411}{629}\right)\) | \(e\left(\frac{1051}{1258}\right)\) | \(e\left(\frac{353}{1258}\right)\) |
\(\chi_{1259}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1153}{1258}\right)\) | \(e\left(\frac{328}{629}\right)\) | \(e\left(\frac{524}{629}\right)\) | \(e\left(\frac{413}{629}\right)\) | \(e\left(\frac{551}{1258}\right)\) | \(e\left(\frac{294}{629}\right)\) | \(e\left(\frac{943}{1258}\right)\) | \(e\left(\frac{27}{629}\right)\) | \(e\left(\frac{721}{1258}\right)\) | \(e\left(\frac{1171}{1258}\right)\) |
\(\chi_{1259}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1129}{1258}\right)\) | \(e\left(\frac{385}{629}\right)\) | \(e\left(\frac{500}{629}\right)\) | \(e\left(\frac{130}{629}\right)\) | \(e\left(\frac{641}{1258}\right)\) | \(e\left(\frac{487}{629}\right)\) | \(e\left(\frac{871}{1258}\right)\) | \(e\left(\frac{141}{629}\right)\) | \(e\left(\frac{131}{1258}\right)\) | \(e\left(\frac{1223}{1258}\right)\) |
\(\chi_{1259}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{1258}\right)\) | \(e\left(\frac{314}{629}\right)\) | \(e\left(\frac{563}{629}\right)\) | \(e\left(\frac{8}{629}\right)\) | \(e\left(\frac{1191}{1258}\right)\) | \(e\left(\frac{59}{629}\right)\) | \(e\left(\frac{431}{1258}\right)\) | \(e\left(\frac{628}{629}\right)\) | \(e\left(\frac{579}{1258}\right)\) | \(e\left(\frac{143}{1258}\right)\) |
\(\chi_{1259}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{681}{1258}\right)\) | \(e\left(\frac{191}{629}\right)\) | \(e\left(\frac{52}{629}\right)\) | \(e\left(\frac{89}{629}\right)\) | \(e\left(\frac{1063}{1258}\right)\) | \(e\left(\frac{106}{629}\right)\) | \(e\left(\frac{785}{1258}\right)\) | \(e\left(\frac{382}{629}\right)\) | \(e\left(\frac{859}{1258}\right)\) | \(e\left(\frac{97}{1258}\right)\) |
\(\chi_{1259}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{773}{1258}\right)\) | \(e\left(\frac{287}{629}\right)\) | \(e\left(\frac{144}{629}\right)\) | \(e\left(\frac{440}{629}\right)\) | \(e\left(\frac{89}{1258}\right)\) | \(e\left(\frac{100}{629}\right)\) | \(e\left(\frac{1061}{1258}\right)\) | \(e\left(\frac{574}{629}\right)\) | \(e\left(\frac{395}{1258}\right)\) | \(e\left(\frac{317}{1258}\right)\) |
\(\chi_{1259}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1227}{1258}\right)\) | \(e\left(\frac{624}{629}\right)\) | \(e\left(\frac{598}{629}\right)\) | \(e\left(\frac{80}{629}\right)\) | \(e\left(\frac{1217}{1258}\right)\) | \(e\left(\frac{590}{629}\right)\) | \(e\left(\frac{1165}{1258}\right)\) | \(e\left(\frac{619}{629}\right)\) | \(e\left(\frac{129}{1258}\right)\) | \(e\left(\frac{801}{1258}\right)\) |
\(\chi_{1259}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{249}{1258}\right)\) | \(e\left(\frac{588}{629}\right)\) | \(e\left(\frac{249}{629}\right)\) | \(e\left(\frac{27}{629}\right)\) | \(e\left(\frac{167}{1258}\right)\) | \(e\left(\frac{435}{629}\right)\) | \(e\left(\frac{747}{1258}\right)\) | \(e\left(\frac{547}{629}\right)\) | \(e\left(\frac{303}{1258}\right)\) | \(e\left(\frac{1033}{1258}\right)\) |
\(\chi_{1259}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{811}{1258}\right)\) | \(e\left(\frac{354}{629}\right)\) | \(e\left(\frac{182}{629}\right)\) | \(e\left(\frac{626}{629}\right)\) | \(e\left(\frac{261}{1258}\right)\) | \(e\left(\frac{371}{629}\right)\) | \(e\left(\frac{1175}{1258}\right)\) | \(e\left(\frac{79}{629}\right)\) | \(e\left(\frac{805}{1258}\right)\) | \(e\left(\frac{25}{1258}\right)\) |
\(\chi_{1259}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{623}{1258}\right)\) | \(e\left(\frac{486}{629}\right)\) | \(e\left(\frac{623}{629}\right)\) | \(e\left(\frac{401}{629}\right)\) | \(e\left(\frac{337}{1258}\right)\) | \(e\left(\frac{520}{629}\right)\) | \(e\left(\frac{611}{1258}\right)\) | \(e\left(\frac{343}{629}\right)\) | \(e\left(\frac{167}{1258}\right)\) | \(e\left(\frac{13}{1258}\right)\) |
\(\chi_{1259}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{755}{1258}\right)\) | \(e\left(\frac{487}{629}\right)\) | \(e\left(\frac{126}{629}\right)\) | \(e\left(\frac{385}{629}\right)\) | \(e\left(\frac{471}{1258}\right)\) | \(e\left(\frac{402}{629}\right)\) | \(e\left(\frac{1007}{1258}\right)\) | \(e\left(\frac{345}{629}\right)\) | \(e\left(\frac{267}{1258}\right)\) | \(e\left(\frac{985}{1258}\right)\) |
\(\chi_{1259}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{937}{1258}\right)\) | \(e\left(\frac{212}{629}\right)\) | \(e\left(\frac{308}{629}\right)\) | \(e\left(\frac{382}{629}\right)\) | \(e\left(\frac{103}{1258}\right)\) | \(e\left(\frac{144}{629}\right)\) | \(e\left(\frac{295}{1258}\right)\) | \(e\left(\frac{424}{629}\right)\) | \(e\left(\frac{443}{1258}\right)\) | \(e\left(\frac{381}{1258}\right)\) |
\(\chi_{1259}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{615}{1258}\right)\) | \(e\left(\frac{505}{629}\right)\) | \(e\left(\frac{615}{629}\right)\) | \(e\left(\frac{97}{629}\right)\) | \(e\left(\frac{367}{1258}\right)\) | \(e\left(\frac{165}{629}\right)\) | \(e\left(\frac{587}{1258}\right)\) | \(e\left(\frac{381}{629}\right)\) | \(e\left(\frac{809}{1258}\right)\) | \(e\left(\frac{869}{1258}\right)\) |
\(\chi_{1259}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{975}{1258}\right)\) | \(e\left(\frac{279}{629}\right)\) | \(e\left(\frac{346}{629}\right)\) | \(e\left(\frac{568}{629}\right)\) | \(e\left(\frac{275}{1258}\right)\) | \(e\left(\frac{415}{629}\right)\) | \(e\left(\frac{409}{1258}\right)\) | \(e\left(\frac{558}{629}\right)\) | \(e\left(\frac{853}{1258}\right)\) | \(e\left(\frac{89}{1258}\right)\) |
\(\chi_{1259}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{1258}\right)\) | \(e\left(\frac{63}{629}\right)\) | \(e\left(\frac{139}{629}\right)\) | \(e\left(\frac{250}{629}\right)\) | \(e\left(\frac{265}{1258}\right)\) | \(e\left(\frac{114}{629}\right)\) | \(e\left(\frac{417}{1258}\right)\) | \(e\left(\frac{126}{629}\right)\) | \(e\left(\frac{639}{1258}\right)\) | \(e\left(\frac{223}{1258}\right)\) |
\(\chi_{1259}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{351}{1258}\right)\) | \(e\left(\frac{503}{629}\right)\) | \(e\left(\frac{351}{629}\right)\) | \(e\left(\frac{129}{629}\right)\) | \(e\left(\frac{99}{1258}\right)\) | \(e\left(\frac{401}{629}\right)\) | \(e\left(\frac{1053}{1258}\right)\) | \(e\left(\frac{377}{629}\right)\) | \(e\left(\frac{609}{1258}\right)\) | \(e\left(\frac{183}{1258}\right)\) |
\(\chi_{1259}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1215}{1258}\right)\) | \(e\left(\frac{338}{629}\right)\) | \(e\left(\frac{586}{629}\right)\) | \(e\left(\frac{253}{629}\right)\) | \(e\left(\frac{633}{1258}\right)\) | \(e\left(\frac{372}{629}\right)\) | \(e\left(\frac{1129}{1258}\right)\) | \(e\left(\frac{47}{629}\right)\) | \(e\left(\frac{463}{1258}\right)\) | \(e\left(\frac{827}{1258}\right)\) |