from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1193, base_ring=CyclotomicField(1192))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1193))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1193\) | |
| Conductor: | \(1193\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1192\) | 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: | $\Q(\zeta_{1192})$ |
| Fixed field: | Number field defined by a degree 1192 polynomial (not computed) |
First 31 of 592 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1193}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{298}\right)\) | \(e\left(\frac{1}{1192}\right)\) | \(e\left(\frac{147}{149}\right)\) | \(e\left(\frac{699}{1192}\right)\) | \(e\left(\frac{589}{1192}\right)\) | \(e\left(\frac{175}{1192}\right)\) | \(e\left(\frac{143}{298}\right)\) | \(e\left(\frac{1}{596}\right)\) | \(e\left(\frac{95}{1192}\right)\) | \(e\left(\frac{29}{298}\right)\) |
| \(\chi_{1193}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{298}\right)\) | \(e\left(\frac{699}{1192}\right)\) | \(e\left(\frac{92}{149}\right)\) | \(e\left(\frac{1073}{1192}\right)\) | \(e\left(\frac{471}{1192}\right)\) | \(e\left(\frac{741}{1192}\right)\) | \(e\left(\frac{127}{298}\right)\) | \(e\left(\frac{103}{596}\right)\) | \(e\left(\frac{845}{1192}\right)\) | \(e\left(\frac{7}{298}\right)\) |
| \(\chi_{1193}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{298}\right)\) | \(e\left(\frac{589}{1192}\right)\) | \(e\left(\frac{14}{149}\right)\) | \(e\left(\frac{471}{1192}\right)\) | \(e\left(\frac{49}{1192}\right)\) | \(e\left(\frac{563}{1192}\right)\) | \(e\left(\frac{191}{298}\right)\) | \(e\left(\frac{589}{596}\right)\) | \(e\left(\frac{1123}{1192}\right)\) | \(e\left(\frac{95}{298}\right)\) |
| \(\chi_{1193}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{298}\right)\) | \(e\left(\frac{175}{1192}\right)\) | \(e\left(\frac{97}{149}\right)\) | \(e\left(\frac{741}{1192}\right)\) | \(e\left(\frac{563}{1192}\right)\) | \(e\left(\frac{825}{1192}\right)\) | \(e\left(\frac{291}{298}\right)\) | \(e\left(\frac{175}{596}\right)\) | \(e\left(\frac{1129}{1192}\right)\) | \(e\left(\frac{9}{298}\right)\) |
| \(\chi_{1193}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{298}\right)\) | \(e\left(\frac{95}{1192}\right)\) | \(e\left(\frac{108}{149}\right)\) | \(e\left(\frac{845}{1192}\right)\) | \(e\left(\frac{1123}{1192}\right)\) | \(e\left(\frac{1129}{1192}\right)\) | \(e\left(\frac{175}{298}\right)\) | \(e\left(\frac{95}{596}\right)\) | \(e\left(\frac{681}{1192}\right)\) | \(e\left(\frac{73}{298}\right)\) |
| \(\chi_{1193}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{298}\right)\) | \(e\left(\frac{1177}{1192}\right)\) | \(e\left(\frac{30}{149}\right)\) | \(e\left(\frac{243}{1192}\right)\) | \(e\left(\frac{701}{1192}\right)\) | \(e\left(\frac{951}{1192}\right)\) | \(e\left(\frac{239}{298}\right)\) | \(e\left(\frac{581}{596}\right)\) | \(e\left(\frac{959}{1192}\right)\) | \(e\left(\frac{161}{298}\right)\) |
| \(\chi_{1193}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{298}\right)\) | \(e\left(\frac{763}{1192}\right)\) | \(e\left(\frac{113}{149}\right)\) | \(e\left(\frac{513}{1192}\right)\) | \(e\left(\frac{23}{1192}\right)\) | \(e\left(\frac{21}{1192}\right)\) | \(e\left(\frac{41}{298}\right)\) | \(e\left(\frac{167}{596}\right)\) | \(e\left(\frac{965}{1192}\right)\) | \(e\left(\frac{75}{298}\right)\) |
| \(\chi_{1193}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{298}\right)\) | \(e\left(\frac{327}{1192}\right)\) | \(e\left(\frac{91}{149}\right)\) | \(e\left(\frac{901}{1192}\right)\) | \(e\left(\frac{691}{1192}\right)\) | \(e\left(\frac{9}{1192}\right)\) | \(e\left(\frac{273}{298}\right)\) | \(e\left(\frac{327}{596}\right)\) | \(e\left(\frac{73}{1192}\right)\) | \(e\left(\frac{245}{298}\right)\) |
| \(\chi_{1193}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{237}{298}\right)\) | \(e\left(\frac{999}{1192}\right)\) | \(e\left(\frac{88}{149}\right)\) | \(e\left(\frac{981}{1192}\right)\) | \(e\left(\frac{755}{1192}\right)\) | \(e\left(\frac{793}{1192}\right)\) | \(e\left(\frac{115}{298}\right)\) | \(e\left(\frac{403}{596}\right)\) | \(e\left(\frac{737}{1192}\right)\) | \(e\left(\frac{65}{298}\right)\) |
| \(\chi_{1193}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{273}{298}\right)\) | \(e\left(\frac{683}{1192}\right)\) | \(e\left(\frac{124}{149}\right)\) | \(e\left(\frac{617}{1192}\right)\) | \(e\left(\frac{583}{1192}\right)\) | \(e\left(\frac{325}{1192}\right)\) | \(e\left(\frac{223}{298}\right)\) | \(e\left(\frac{87}{596}\right)\) | \(e\left(\frac{517}{1192}\right)\) | \(e\left(\frac{139}{298}\right)\) |
| \(\chi_{1193}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{298}\right)\) | \(e\left(\frac{951}{1192}\right)\) | \(e\left(\frac{35}{149}\right)\) | \(e\left(\frac{805}{1192}\right)\) | \(e\left(\frac{1091}{1192}\right)\) | \(e\left(\frac{737}{1192}\right)\) | \(e\left(\frac{105}{298}\right)\) | \(e\left(\frac{355}{596}\right)\) | \(e\left(\frac{945}{1192}\right)\) | \(e\left(\frac{163}{298}\right)\) |
| \(\chi_{1193}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{195}{298}\right)\) | \(e\left(\frac{573}{1192}\right)\) | \(e\left(\frac{46}{149}\right)\) | \(e\left(\frac{15}{1192}\right)\) | \(e\left(\frac{161}{1192}\right)\) | \(e\left(\frac{147}{1192}\right)\) | \(e\left(\frac{287}{298}\right)\) | \(e\left(\frac{573}{596}\right)\) | \(e\left(\frac{795}{1192}\right)\) | \(e\left(\frac{227}{298}\right)\) |
| \(\chi_{1193}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{298}\right)\) | \(e\left(\frac{3}{1192}\right)\) | \(e\left(\frac{143}{149}\right)\) | \(e\left(\frac{905}{1192}\right)\) | \(e\left(\frac{575}{1192}\right)\) | \(e\left(\frac{525}{1192}\right)\) | \(e\left(\frac{131}{298}\right)\) | \(e\left(\frac{3}{596}\right)\) | \(e\left(\frac{285}{1192}\right)\) | \(e\left(\frac{87}{298}\right)\) |
| \(\chi_{1193}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{298}\right)\) | \(e\left(\frac{159}{1192}\right)\) | \(e\left(\frac{129}{149}\right)\) | \(e\left(\frac{285}{1192}\right)\) | \(e\left(\frac{675}{1192}\right)\) | \(e\left(\frac{409}{1192}\right)\) | \(e\left(\frac{89}{298}\right)\) | \(e\left(\frac{159}{596}\right)\) | \(e\left(\frac{801}{1192}\right)\) | \(e\left(\frac{141}{298}\right)\) |
| \(\chi_{1193}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{298}\right)\) | \(e\left(\frac{901}{1192}\right)\) | \(e\left(\frac{135}{149}\right)\) | \(e\left(\frac{423}{1192}\right)\) | \(e\left(\frac{249}{1192}\right)\) | \(e\left(\frac{331}{1192}\right)\) | \(e\left(\frac{107}{298}\right)\) | \(e\left(\frac{305}{596}\right)\) | \(e\left(\frac{963}{1192}\right)\) | \(e\left(\frac{203}{298}\right)\) |
| \(\chi_{1193}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{298}\right)\) | \(e\left(\frac{117}{1192}\right)\) | \(e\left(\frac{64}{149}\right)\) | \(e\left(\frac{727}{1192}\right)\) | \(e\left(\frac{969}{1192}\right)\) | \(e\left(\frac{211}{1192}\right)\) | \(e\left(\frac{43}{298}\right)\) | \(e\left(\frac{117}{596}\right)\) | \(e\left(\frac{387}{1192}\right)\) | \(e\left(\frac{115}{298}\right)\) |
| \(\chi_{1193}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{298}\right)\) | \(e\left(\frac{915}{1192}\right)\) | \(e\left(\frac{107}{149}\right)\) | \(e\left(\frac{673}{1192}\right)\) | \(e\left(\frac{151}{1192}\right)\) | \(e\left(\frac{397}{1192}\right)\) | \(e\left(\frac{23}{298}\right)\) | \(e\left(\frac{319}{596}\right)\) | \(e\left(\frac{1101}{1192}\right)\) | \(e\left(\frac{13}{298}\right)\) |
| \(\chi_{1193}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{298}\right)\) | \(e\left(\frac{395}{1192}\right)\) | \(e\left(\frac{104}{149}\right)\) | \(e\left(\frac{753}{1192}\right)\) | \(e\left(\frac{215}{1192}\right)\) | \(e\left(\frac{1181}{1192}\right)\) | \(e\left(\frac{163}{298}\right)\) | \(e\left(\frac{395}{596}\right)\) | \(e\left(\frac{573}{1192}\right)\) | \(e\left(\frac{131}{298}\right)\) |
| \(\chi_{1193}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{298}\right)\) | \(e\left(\frac{259}{1192}\right)\) | \(e\left(\frac{78}{149}\right)\) | \(e\left(\frac{1049}{1192}\right)\) | \(e\left(\frac{1167}{1192}\right)\) | \(e\left(\frac{29}{1192}\right)\) | \(e\left(\frac{85}{298}\right)\) | \(e\left(\frac{259}{596}\right)\) | \(e\left(\frac{765}{1192}\right)\) | \(e\left(\frac{61}{298}\right)\) |
| \(\chi_{1193}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{298}\right)\) | \(e\left(\frac{79}{1192}\right)\) | \(e\left(\frac{140}{149}\right)\) | \(e\left(\frac{389}{1192}\right)\) | \(e\left(\frac{43}{1192}\right)\) | \(e\left(\frac{713}{1192}\right)\) | \(e\left(\frac{271}{298}\right)\) | \(e\left(\frac{79}{596}\right)\) | \(e\left(\frac{353}{1192}\right)\) | \(e\left(\frac{205}{298}\right)\) |
| \(\chi_{1193}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{298}\right)\) | \(e\left(\frac{1149}{1192}\right)\) | \(e\left(\frac{86}{149}\right)\) | \(e\left(\frac{935}{1192}\right)\) | \(e\left(\frac{897}{1192}\right)\) | \(e\left(\frac{819}{1192}\right)\) | \(e\left(\frac{109}{298}\right)\) | \(e\left(\frac{553}{596}\right)\) | \(e\left(\frac{683}{1192}\right)\) | \(e\left(\frac{243}{298}\right)\) |
| \(\chi_{1193}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{237}{298}\right)\) | \(e\left(\frac{701}{1192}\right)\) | \(e\left(\frac{88}{149}\right)\) | \(e\left(\frac{87}{1192}\right)\) | \(e\left(\frac{457}{1192}\right)\) | \(e\left(\frac{1091}{1192}\right)\) | \(e\left(\frac{115}{298}\right)\) | \(e\left(\frac{105}{596}\right)\) | \(e\left(\frac{1035}{1192}\right)\) | \(e\left(\frac{65}{298}\right)\) |
| \(\chi_{1193}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{298}\right)\) | \(e\left(\frac{347}{1192}\right)\) | \(e\left(\frac{51}{149}\right)\) | \(e\left(\frac{577}{1192}\right)\) | \(e\left(\frac{551}{1192}\right)\) | \(e\left(\frac{1125}{1192}\right)\) | \(e\left(\frac{153}{298}\right)\) | \(e\left(\frac{347}{596}\right)\) | \(e\left(\frac{781}{1192}\right)\) | \(e\left(\frac{229}{298}\right)\) |
| \(\chi_{1193}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{298}\right)\) | \(e\left(\frac{1161}{1192}\right)\) | \(e\left(\frac{62}{149}\right)\) | \(e\left(\frac{979}{1192}\right)\) | \(e\left(\frac{813}{1192}\right)\) | \(e\left(\frac{535}{1192}\right)\) | \(e\left(\frac{37}{298}\right)\) | \(e\left(\frac{565}{596}\right)\) | \(e\left(\frac{631}{1192}\right)\) | \(e\left(\frac{293}{298}\right)\) |
| \(\chi_{1193}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{298}\right)\) | \(e\left(\frac{1087}{1192}\right)\) | \(e\left(\frac{61}{149}\right)\) | \(e\left(\frac{509}{1192}\right)\) | \(e\left(\frac{139}{1192}\right)\) | \(e\left(\frac{697}{1192}\right)\) | \(e\left(\frac{183}{298}\right)\) | \(e\left(\frac{491}{596}\right)\) | \(e\left(\frac{753}{1192}\right)\) | \(e\left(\frac{233}{298}\right)\) |
| \(\chi_{1193}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{298}\right)\) | \(e\left(\frac{591}{1192}\right)\) | \(e\left(\frac{10}{149}\right)\) | \(e\left(\frac{677}{1192}\right)\) | \(e\left(\frac{35}{1192}\right)\) | \(e\left(\frac{913}{1192}\right)\) | \(e\left(\frac{179}{298}\right)\) | \(e\left(\frac{591}{596}\right)\) | \(e\left(\frac{121}{1192}\right)\) | \(e\left(\frac{153}{298}\right)\) |
| \(\chi_{1193}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{298}\right)\) | \(e\left(\frac{815}{1192}\right)\) | \(e\left(\frac{9}{149}\right)\) | \(e\left(\frac{1101}{1192}\right)\) | \(e\left(\frac{851}{1192}\right)\) | \(e\left(\frac{777}{1192}\right)\) | \(e\left(\frac{27}{298}\right)\) | \(e\left(\frac{219}{596}\right)\) | \(e\left(\frac{1137}{1192}\right)\) | \(e\left(\frac{93}{298}\right)\) |
| \(\chi_{1193}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{298}\right)\) | \(e\left(\frac{747}{1192}\right)\) | \(e\left(\frac{145}{149}\right)\) | \(e\left(\frac{57}{1192}\right)\) | \(e\left(\frac{135}{1192}\right)\) | \(e\left(\frac{797}{1192}\right)\) | \(e\left(\frac{137}{298}\right)\) | \(e\left(\frac{151}{596}\right)\) | \(e\left(\frac{637}{1192}\right)\) | \(e\left(\frac{207}{298}\right)\) |
| \(\chi_{1193}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{298}\right)\) | \(e\left(\frac{171}{1192}\right)\) | \(e\left(\frac{105}{149}\right)\) | \(e\left(\frac{329}{1192}\right)\) | \(e\left(\frac{591}{1192}\right)\) | \(e\left(\frac{125}{1192}\right)\) | \(e\left(\frac{17}{298}\right)\) | \(e\left(\frac{171}{596}\right)\) | \(e\left(\frac{749}{1192}\right)\) | \(e\left(\frac{191}{298}\right)\) |
| \(\chi_{1193}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{298}\right)\) | \(e\left(\frac{297}{1192}\right)\) | \(e\left(\frac{2}{149}\right)\) | \(e\left(\frac{195}{1192}\right)\) | \(e\left(\frac{901}{1192}\right)\) | \(e\left(\frac{719}{1192}\right)\) | \(e\left(\frac{155}{298}\right)\) | \(e\left(\frac{297}{596}\right)\) | \(e\left(\frac{799}{1192}\right)\) | \(e\left(\frac{269}{298}\right)\) |
| \(\chi_{1193}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{298}\right)\) | \(e\left(\frac{177}{1192}\right)\) | \(e\left(\frac{93}{149}\right)\) | \(e\left(\frac{947}{1192}\right)\) | \(e\left(\frac{549}{1192}\right)\) | \(e\left(\frac{1175}{1192}\right)\) | \(e\left(\frac{279}{298}\right)\) | \(e\left(\frac{177}{596}\right)\) | \(e\left(\frac{127}{1192}\right)\) | \(e\left(\frac{67}{298}\right)\) |