from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(924))
M = H._module
chi = DirichletCharacter(H, M([308,252,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,6003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6003\) | |
| Conductor: | \(6003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(924\) | 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: | $\Q(\zeta_{924})$ |
| Fixed field: | Number field defined by a degree 924 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6003}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{845}{924}\right)\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{335}{462}\right)\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{229}{308}\right)\) | \(e\left(\frac{197}{308}\right)\) | \(e\left(\frac{629}{924}\right)\) | \(e\left(\frac{59}{462}\right)\) | \(e\left(\frac{793}{924}\right)\) | \(e\left(\frac{152}{231}\right)\) |
| \(\chi_{6003}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{551}{924}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{243}{308}\right)\) | \(e\left(\frac{127}{308}\right)\) | \(e\left(\frac{923}{924}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{247}{924}\right)\) | \(e\left(\frac{89}{231}\right)\) |
| \(\chi_{6003}(124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{631}{924}\right)\) | \(e\left(\frac{169}{462}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{136}{231}\right)\) | \(e\left(\frac{15}{308}\right)\) | \(e\left(\frac{255}{308}\right)\) | \(e\left(\frac{403}{924}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{251}{924}\right)\) | \(e\left(\frac{169}{231}\right)\) |
| \(\chi_{6003}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{655}{924}\right)\) | \(e\left(\frac{193}{462}\right)\) | \(e\left(\frac{205}{462}\right)\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{39}{308}\right)\) | \(e\left(\frac{47}{308}\right)\) | \(e\left(\frac{247}{924}\right)\) | \(e\left(\frac{43}{462}\right)\) | \(e\left(\frac{899}{924}\right)\) | \(e\left(\frac{193}{231}\right)\) |
| \(\chi_{6003}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{521}{924}\right)\) | \(e\left(\frac{59}{462}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{191}{231}\right)\) | \(e\left(\frac{213}{308}\right)\) | \(e\left(\frac{233}{308}\right)\) | \(e\left(\frac{425}{924}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{361}{924}\right)\) | \(e\left(\frac{59}{231}\right)\) |
| \(\chi_{6003}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{743}{924}\right)\) | \(e\left(\frac{281}{462}\right)\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{127}{308}\right)\) | \(e\left(\frac{3}{308}\right)\) | \(e\left(\frac{599}{924}\right)\) | \(e\left(\frac{65}{462}\right)\) | \(e\left(\frac{811}{924}\right)\) | \(e\left(\frac{50}{231}\right)\) |
| \(\chi_{6003}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{775}{924}\right)\) | \(e\left(\frac{313}{462}\right)\) | \(e\left(\frac{433}{462}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{159}{308}\right)\) | \(e\left(\frac{239}{308}\right)\) | \(e\left(\frac{391}{924}\right)\) | \(e\left(\frac{199}{462}\right)\) | \(e\left(\frac{443}{924}\right)\) | \(e\left(\frac{82}{231}\right)\) |
| \(\chi_{6003}(292,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{924}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{89}{308}\right)\) | \(e\left(\frac{281}{308}\right)\) | \(e\left(\frac{461}{924}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{709}{924}\right)\) | \(e\left(\frac{89}{231}\right)\) |
| \(\chi_{6003}(301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{924}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{69}{308}\right)\) | \(e\left(\frac{249}{308}\right)\) | \(e\left(\frac{437}{924}\right)\) | \(e\left(\frac{5}{462}\right)\) | \(e\left(\frac{169}{924}\right)\) | \(e\left(\frac{146}{231}\right)\) |
| \(\chi_{6003}(340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{924}\right)\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{109}{462}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{29}{308}\right)\) | \(e\left(\frac{185}{308}\right)\) | \(e\left(\frac{697}{924}\right)\) | \(e\left(\frac{415}{462}\right)\) | \(e\left(\frac{629}{924}\right)\) | \(e\left(\frac{106}{231}\right)\) |
| \(\chi_{6003}(358,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{703}{924}\right)\) | \(e\left(\frac{241}{462}\right)\) | \(e\left(\frac{19}{462}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{87}{308}\right)\) | \(e\left(\frac{247}{308}\right)\) | \(e\left(\frac{859}{924}\right)\) | \(e\left(\frac{13}{462}\right)\) | \(e\left(\frac{347}{924}\right)\) | \(e\left(\frac{10}{231}\right)\) |
| \(\chi_{6003}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{924}\right)\) | \(e\left(\frac{151}{462}\right)\) | \(e\left(\frac{79}{462}\right)\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{151}{308}\right)\) | \(e\left(\frac{103}{308}\right)\) | \(e\left(\frac{751}{924}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{227}{924}\right)\) | \(e\left(\frac{151}{231}\right)\) |
| \(\chi_{6003}(409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{557}{924}\right)\) | \(e\left(\frac{95}{462}\right)\) | \(e\left(\frac{65}{462}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{249}{308}\right)\) | \(e\left(\frac{229}{308}\right)\) | \(e\left(\frac{653}{924}\right)\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{409}{924}\right)\) | \(e\left(\frac{95}{231}\right)\) |
| \(\chi_{6003}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{924}\right)\) | \(e\left(\frac{197}{462}\right)\) | \(e\left(\frac{305}{462}\right)\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{197}{308}\right)\) | \(e\left(\frac{269}{308}\right)\) | \(e\left(\frac{221}{924}\right)\) | \(e\left(\frac{233}{462}\right)\) | \(e\left(\frac{853}{924}\right)\) | \(e\left(\frac{197}{231}\right)\) |
| \(\chi_{6003}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{647}{924}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{5}{462}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{31}{308}\right)\) | \(e\left(\frac{219}{308}\right)\) | \(e\left(\frac{299}{924}\right)\) | \(e\left(\frac{125}{462}\right)\) | \(e\left(\frac{67}{924}\right)\) | \(e\left(\frac{185}{231}\right)\) |
| \(\chi_{6003}(466,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{924}\right)\) | \(e\left(\frac{313}{462}\right)\) | \(e\left(\frac{433}{462}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{5}{308}\right)\) | \(e\left(\frac{85}{308}\right)\) | \(e\left(\frac{853}{924}\right)\) | \(e\left(\frac{199}{462}\right)\) | \(e\left(\frac{905}{924}\right)\) | \(e\left(\frac{82}{231}\right)\) |
| \(\chi_{6003}(472,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{924}\right)\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{239}{308}\right)\) | \(e\left(\frac{59}{308}\right)\) | \(e\left(\frac{179}{924}\right)\) | \(e\left(\frac{149}{462}\right)\) | \(e\left(\frac{139}{924}\right)\) | \(e\left(\frac{8}{231}\right)\) |
| \(\chi_{6003}(508,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{924}\right)\) | \(e\left(\frac{443}{462}\right)\) | \(e\left(\frac{449}{462}\right)\) | \(e\left(\frac{146}{231}\right)\) | \(e\left(\frac{135}{308}\right)\) | \(e\left(\frac{139}{308}\right)\) | \(e\left(\frac{239}{924}\right)\) | \(e\left(\frac{137}{462}\right)\) | \(e\left(\frac{103}{924}\right)\) | \(e\left(\frac{212}{231}\right)\) |
| \(\chi_{6003}(565,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{924}\right)\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{65}{308}\right)\) | \(e\left(\frac{181}{308}\right)\) | \(e\left(\frac{1}{924}\right)\) | \(e\left(\frac{277}{462}\right)\) | \(e\left(\frac{677}{924}\right)\) | \(e\left(\frac{142}{231}\right)\) |
| \(\chi_{6003}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{924}\right)\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{247}{462}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{53}{308}\right)\) | \(e\left(\frac{285}{308}\right)\) | \(e\left(\frac{541}{924}\right)\) | \(e\left(\frac{169}{462}\right)\) | \(e\left(\frac{353}{924}\right)\) | \(e\left(\frac{130}{231}\right)\) |
| \(\chi_{6003}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{673}{924}\right)\) | \(e\left(\frac{211}{462}\right)\) | \(e\left(\frac{193}{462}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{57}{308}\right)\) | \(e\left(\frac{45}{308}\right)\) | \(e\left(\frac{361}{924}\right)\) | \(e\left(\frac{205}{462}\right)\) | \(e\left(\frac{461}{924}\right)\) | \(e\left(\frac{211}{231}\right)\) |
| \(\chi_{6003}(607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{719}{924}\right)\) | \(e\left(\frac{257}{462}\right)\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{103}{308}\right)\) | \(e\left(\frac{211}{308}\right)\) | \(e\left(\frac{755}{924}\right)\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{163}{924}\right)\) | \(e\left(\frac{26}{231}\right)\) |
| \(\chi_{6003}(646,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{883}{924}\right)\) | \(e\left(\frac{421}{462}\right)\) | \(e\left(\frac{361}{462}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{267}{308}\right)\) | \(e\left(\frac{227}{308}\right)\) | \(e\left(\frac{151}{924}\right)\) | \(e\left(\frac{247}{462}\right)\) | \(e\left(\frac{587}{924}\right)\) | \(e\left(\frac{190}{231}\right)\) |
| \(\chi_{6003}(652,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{924}\right)\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{71}{462}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{9}{308}\right)\) | \(e\left(\frac{153}{308}\right)\) | \(e\left(\frac{365}{924}\right)\) | \(e\left(\frac{389}{462}\right)\) | \(e\left(\frac{397}{924}\right)\) | \(e\left(\frac{86}{231}\right)\) |
| \(\chi_{6003}(670,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{893}{924}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{149}{462}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{277}{308}\right)\) | \(e\left(\frac{89}{308}\right)\) | \(e\left(\frac{317}{924}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{241}{924}\right)\) | \(e\left(\frac{200}{231}\right)\) |
| \(\chi_{6003}(706,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{815}{924}\right)\) | \(e\left(\frac{353}{462}\right)\) | \(e\left(\frac{47}{462}\right)\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{199}{308}\right)\) | \(e\left(\frac{303}{308}\right)\) | \(e\left(\frac{131}{924}\right)\) | \(e\left(\frac{251}{462}\right)\) | \(e\left(\frac{907}{924}\right)\) | \(e\left(\frac{122}{231}\right)\) |
| \(\chi_{6003}(715,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{773}{924}\right)\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{383}{462}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{157}{308}\right)\) | \(e\left(\frac{205}{308}\right)\) | \(e\left(\frac{173}{924}\right)\) | \(e\left(\frac{335}{462}\right)\) | \(e\left(\frac{697}{924}\right)\) | \(e\left(\frac{80}{231}\right)\) |
| \(\chi_{6003}(772,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{924}\right)\) | \(e\left(\frac{307}{462}\right)\) | \(e\left(\frac{283}{462}\right)\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{307}{308}\right)\) | \(e\left(\frac{291}{308}\right)\) | \(e\left(\frac{199}{924}\right)\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{743}{924}\right)\) | \(e\left(\frac{76}{231}\right)\) |
| \(\chi_{6003}(814,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{924}\right)\) | \(e\left(\frac{257}{462}\right)\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{257}{308}\right)\) | \(e\left(\frac{57}{308}\right)\) | \(e\left(\frac{293}{924}\right)\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{625}{924}\right)\) | \(e\left(\frac{26}{231}\right)\) |
| \(\chi_{6003}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{924}\right)\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{395}{462}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{293}{308}\right)\) | \(e\left(\frac{53}{308}\right)\) | \(e\left(\frac{521}{924}\right)\) | \(e\left(\frac{173}{462}\right)\) | \(e\left(\frac{673}{924}\right)\) | \(e\left(\frac{62}{231}\right)\) |
| \(\chi_{6003}(844,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{529}{924}\right)\) | \(e\left(\frac{67}{462}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{166}{231}\right)\) | \(e\left(\frac{221}{308}\right)\) | \(e\left(\frac{61}{308}\right)\) | \(e\left(\frac{373}{924}\right)\) | \(e\left(\frac{295}{462}\right)\) | \(e\left(\frac{269}{924}\right)\) | \(e\left(\frac{67}{231}\right)\) |