from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1303, base_ring=CyclotomicField(1302))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,1303))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1303\) | |
| Conductor: | \(1303\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1302\) | 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: | $\Q(\zeta_{651})$ |
| Fixed field: | Number field defined by a degree 1302 polynomial (not computed) |
First 31 of 360 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1303}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{651}\right)\) | \(e\left(\frac{127}{434}\right)\) | \(e\left(\frac{271}{651}\right)\) | \(e\left(\frac{29}{62}\right)\) | \(e\left(\frac{1}{1302}\right)\) | \(e\left(\frac{655}{1302}\right)\) | \(e\left(\frac{27}{217}\right)\) | \(e\left(\frac{127}{217}\right)\) | \(e\left(\frac{229}{1302}\right)\) | \(e\left(\frac{19}{62}\right)\) |
| \(\chi_{1303}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{542}{651}\right)\) | \(e\left(\frac{291}{434}\right)\) | \(e\left(\frac{433}{651}\right)\) | \(e\left(\frac{23}{62}\right)\) | \(e\left(\frac{655}{1302}\right)\) | \(e\left(\frac{667}{1302}\right)\) | \(e\left(\frac{108}{217}\right)\) | \(e\left(\frac{74}{217}\right)\) | \(e\left(\frac{265}{1302}\right)\) | \(e\left(\frac{45}{62}\right)\) |
| \(\chi_{1303}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{651}\right)\) | \(e\left(\frac{5}{434}\right)\) | \(e\left(\frac{214}{651}\right)\) | \(e\left(\frac{7}{62}\right)\) | \(e\left(\frac{229}{1302}\right)\) | \(e\left(\frac{265}{1302}\right)\) | \(e\left(\frac{107}{217}\right)\) | \(e\left(\frac{5}{217}\right)\) | \(e\left(\frac{361}{1302}\right)\) | \(e\left(\frac{11}{62}\right)\) |
| \(\chi_{1303}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{400}{651}\right)\) | \(e\left(\frac{41}{434}\right)\) | \(e\left(\frac{149}{651}\right)\) | \(e\left(\frac{45}{62}\right)\) | \(e\left(\frac{923}{1302}\right)\) | \(e\left(\frac{437}{1302}\right)\) | \(e\left(\frac{183}{217}\right)\) | \(e\left(\frac{41}{217}\right)\) | \(e\left(\frac{443}{1302}\right)\) | \(e\left(\frac{53}{62}\right)\) |
| \(\chi_{1303}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{481}{651}\right)\) | \(e\left(\frac{205}{434}\right)\) | \(e\left(\frac{311}{651}\right)\) | \(e\left(\frac{39}{62}\right)\) | \(e\left(\frac{275}{1302}\right)\) | \(e\left(\frac{449}{1302}\right)\) | \(e\left(\frac{47}{217}\right)\) | \(e\left(\frac{205}{217}\right)\) | \(e\left(\frac{479}{1302}\right)\) | \(e\left(\frac{17}{62}\right)\) |
| \(\chi_{1303}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{651}\right)\) | \(e\left(\frac{353}{434}\right)\) | \(e\left(\frac{92}{651}\right)\) | \(e\left(\frac{23}{62}\right)\) | \(e\left(\frac{1151}{1302}\right)\) | \(e\left(\frac{47}{1302}\right)\) | \(e\left(\frac{46}{217}\right)\) | \(e\left(\frac{136}{217}\right)\) | \(e\left(\frac{575}{1302}\right)\) | \(e\left(\frac{45}{62}\right)\) |
| \(\chi_{1303}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{296}{651}\right)\) | \(e\left(\frac{243}{434}\right)\) | \(e\left(\frac{592}{651}\right)\) | \(e\left(\frac{55}{62}\right)\) | \(e\left(\frac{19}{1302}\right)\) | \(e\left(\frac{727}{1302}\right)\) | \(e\left(\frac{79}{217}\right)\) | \(e\left(\frac{26}{217}\right)\) | \(e\left(\frac{445}{1302}\right)\) | \(e\left(\frac{51}{62}\right)\) |
| \(\chi_{1303}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{651}\right)\) | \(e\left(\frac{265}{434}\right)\) | \(e\left(\frac{58}{651}\right)\) | \(e\left(\frac{61}{62}\right)\) | \(e\left(\frac{853}{1302}\right)\) | \(e\left(\frac{157}{1302}\right)\) | \(e\left(\frac{29}{217}\right)\) | \(e\left(\frac{48}{217}\right)\) | \(e\left(\frac{37}{1302}\right)\) | \(e\left(\frac{25}{62}\right)\) |
| \(\chi_{1303}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{651}\right)\) | \(e\left(\frac{233}{434}\right)\) | \(e\left(\frac{598}{651}\right)\) | \(e\left(\frac{41}{62}\right)\) | \(e\left(\frac{1297}{1302}\right)\) | \(e\left(\frac{631}{1302}\right)\) | \(e\left(\frac{82}{217}\right)\) | \(e\left(\frac{16}{217}\right)\) | \(e\left(\frac{157}{1302}\right)\) | \(e\left(\frac{29}{62}\right)\) |
| \(\chi_{1303}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{152}{651}\right)\) | \(e\left(\frac{289}{434}\right)\) | \(e\left(\frac{304}{651}\right)\) | \(e\left(\frac{45}{62}\right)\) | \(e\left(\frac{1171}{1302}\right)\) | \(e\left(\frac{127}{1302}\right)\) | \(e\left(\frac{152}{217}\right)\) | \(e\left(\frac{72}{217}\right)\) | \(e\left(\frac{1249}{1302}\right)\) | \(e\left(\frac{53}{62}\right)\) |
| \(\chi_{1303}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{651}\right)\) | \(e\left(\frac{11}{434}\right)\) | \(e\left(\frac{167}{651}\right)\) | \(e\left(\frac{3}{62}\right)\) | \(e\left(\frac{851}{1302}\right)\) | \(e\left(\frac{149}{1302}\right)\) | \(e\left(\frac{192}{217}\right)\) | \(e\left(\frac{11}{217}\right)\) | \(e\left(\frac{881}{1302}\right)\) | \(e\left(\frac{49}{62}\right)\) |
| \(\chi_{1303}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{651}\right)\) | \(e\left(\frac{257}{434}\right)\) | \(e\left(\frac{410}{651}\right)\) | \(e\left(\frac{25}{62}\right)\) | \(e\left(\frac{1181}{1302}\right)\) | \(e\left(\frac{167}{1302}\right)\) | \(e\left(\frac{205}{217}\right)\) | \(e\left(\frac{40}{217}\right)\) | \(e\left(\frac{935}{1302}\right)\) | \(e\left(\frac{57}{62}\right)\) |
| \(\chi_{1303}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{651}\right)\) | \(e\left(\frac{199}{434}\right)\) | \(e\left(\frac{358}{651}\right)\) | \(e\left(\frac{43}{62}\right)\) | \(e\left(\frac{955}{1302}\right)\) | \(e\left(\frac{565}{1302}\right)\) | \(e\left(\frac{179}{217}\right)\) | \(e\left(\frac{199}{217}\right)\) | \(e\left(\frac{1261}{1302}\right)\) | \(e\left(\frac{41}{62}\right)\) |
| \(\chi_{1303}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{651}\right)\) | \(e\left(\frac{157}{434}\right)\) | \(e\left(\frac{470}{651}\right)\) | \(e\left(\frac{9}{62}\right)\) | \(e\left(\frac{941}{1302}\right)\) | \(e\left(\frac{509}{1302}\right)\) | \(e\left(\frac{18}{217}\right)\) | \(e\left(\frac{157}{217}\right)\) | \(e\left(\frac{659}{1302}\right)\) | \(e\left(\frac{23}{62}\right)\) |
| \(\chi_{1303}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{278}{651}\right)\) | \(e\left(\frac{303}{434}\right)\) | \(e\left(\frac{556}{651}\right)\) | \(e\left(\frac{15}{62}\right)\) | \(e\left(\frac{163}{1302}\right)\) | \(e\left(\frac{1}{1302}\right)\) | \(e\left(\frac{61}{217}\right)\) | \(e\left(\frac{86}{217}\right)\) | \(e\left(\frac{871}{1302}\right)\) | \(e\left(\frac{59}{62}\right)\) |
| \(\chi_{1303}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{611}{651}\right)\) | \(e\left(\frac{61}{434}\right)\) | \(e\left(\frac{571}{651}\right)\) | \(e\left(\frac{11}{62}\right)\) | \(e\left(\frac{103}{1302}\right)\) | \(e\left(\frac{1063}{1302}\right)\) | \(e\left(\frac{177}{217}\right)\) | \(e\left(\frac{61}{217}\right)\) | \(e\left(\frac{151}{1302}\right)\) | \(e\left(\frac{35}{62}\right)\) |
| \(\chi_{1303}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{651}\right)\) | \(e\left(\frac{33}{434}\right)\) | \(e\left(\frac{67}{651}\right)\) | \(e\left(\frac{9}{62}\right)\) | \(e\left(\frac{817}{1302}\right)\) | \(e\left(\frac{13}{1302}\right)\) | \(e\left(\frac{142}{217}\right)\) | \(e\left(\frac{33}{217}\right)\) | \(e\left(\frac{907}{1302}\right)\) | \(e\left(\frac{23}{62}\right)\) |
| \(\chi_{1303}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{619}{651}\right)\) | \(e\left(\frac{179}{434}\right)\) | \(e\left(\frac{587}{651}\right)\) | \(e\left(\frac{15}{62}\right)\) | \(e\left(\frac{473}{1302}\right)\) | \(e\left(\frac{1241}{1302}\right)\) | \(e\left(\frac{185}{217}\right)\) | \(e\left(\frac{179}{217}\right)\) | \(e\left(\frac{251}{1302}\right)\) | \(e\left(\frac{59}{62}\right)\) |
| \(\chi_{1303}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{523}{651}\right)\) | \(e\left(\frac{65}{434}\right)\) | \(e\left(\frac{395}{651}\right)\) | \(e\left(\frac{29}{62}\right)\) | \(e\left(\frac{1241}{1302}\right)\) | \(e\left(\frac{407}{1302}\right)\) | \(e\left(\frac{89}{217}\right)\) | \(e\left(\frac{65}{217}\right)\) | \(e\left(\frac{353}{1302}\right)\) | \(e\left(\frac{19}{62}\right)\) |
| \(\chi_{1303}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{284}{651}\right)\) | \(e\left(\frac{283}{434}\right)\) | \(e\left(\frac{568}{651}\right)\) | \(e\left(\frac{49}{62}\right)\) | \(e\left(\frac{115}{1302}\right)\) | \(e\left(\frac{1111}{1302}\right)\) | \(e\left(\frac{67}{217}\right)\) | \(e\left(\frac{66}{217}\right)\) | \(e\left(\frac{295}{1302}\right)\) | \(e\left(\frac{15}{62}\right)\) |
| \(\chi_{1303}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{202}{651}\right)\) | \(e\left(\frac{267}{434}\right)\) | \(e\left(\frac{404}{651}\right)\) | \(e\left(\frac{39}{62}\right)\) | \(e\left(\frac{1205}{1302}\right)\) | \(e\left(\frac{263}{1302}\right)\) | \(e\left(\frac{202}{217}\right)\) | \(e\left(\frac{50}{217}\right)\) | \(e\left(\frac{1223}{1302}\right)\) | \(e\left(\frac{17}{62}\right)\) |
| \(\chi_{1303}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{651}\right)\) | \(e\left(\frac{59}{434}\right)\) | \(e\left(\frac{8}{651}\right)\) | \(e\left(\frac{33}{62}\right)\) | \(e\left(\frac{185}{1302}\right)\) | \(e\left(\frac{89}{1302}\right)\) | \(e\left(\frac{4}{217}\right)\) | \(e\left(\frac{59}{217}\right)\) | \(e\left(\frac{701}{1302}\right)\) | \(e\left(\frac{43}{62}\right)\) |
| \(\chi_{1303}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{651}\right)\) | \(e\left(\frac{423}{434}\right)\) | \(e\left(\frac{50}{651}\right)\) | \(e\left(\frac{59}{62}\right)\) | \(e\left(\frac{17}{1302}\right)\) | \(e\left(\frac{719}{1302}\right)\) | \(e\left(\frac{25}{217}\right)\) | \(e\left(\frac{206}{217}\right)\) | \(e\left(\frac{1289}{1302}\right)\) | \(e\left(\frac{13}{62}\right)\) |
| \(\chi_{1303}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{651}\right)\) | \(e\left(\frac{113}{434}\right)\) | \(e\left(\frac{236}{651}\right)\) | \(e\left(\frac{59}{62}\right)\) | \(e\left(\frac{575}{1302}\right)\) | \(e\left(\frac{347}{1302}\right)\) | \(e\left(\frac{118}{217}\right)\) | \(e\left(\frac{113}{217}\right)\) | \(e\left(\frac{173}{1302}\right)\) | \(e\left(\frac{13}{62}\right)\) |
| \(\chi_{1303}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{575}{651}\right)\) | \(e\left(\frac{181}{434}\right)\) | \(e\left(\frac{499}{651}\right)\) | \(e\left(\frac{55}{62}\right)\) | \(e\left(\frac{391}{1302}\right)\) | \(e\left(\frac{913}{1302}\right)\) | \(e\left(\frac{141}{217}\right)\) | \(e\left(\frac{181}{217}\right)\) | \(e\left(\frac{1003}{1302}\right)\) | \(e\left(\frac{51}{62}\right)\) |
| \(\chi_{1303}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{651}\right)\) | \(e\left(\frac{333}{434}\right)\) | \(e\left(\frac{538}{651}\right)\) | \(e\left(\frac{57}{62}\right)\) | \(e\left(\frac{235}{1302}\right)\) | \(e\left(\frac{289}{1302}\right)\) | \(e\left(\frac{52}{217}\right)\) | \(e\left(\frac{116}{217}\right)\) | \(e\left(\frac{433}{1302}\right)\) | \(e\left(\frac{1}{62}\right)\) |
| \(\chi_{1303}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{500}{651}\right)\) | \(e\left(\frac{431}{434}\right)\) | \(e\left(\frac{349}{651}\right)\) | \(e\left(\frac{33}{62}\right)\) | \(e\left(\frac{991}{1302}\right)\) | \(e\left(\frac{709}{1302}\right)\) | \(e\left(\frac{66}{217}\right)\) | \(e\left(\frac{214}{217}\right)\) | \(e\left(\frac{391}{1302}\right)\) | \(e\left(\frac{43}{62}\right)\) |
| \(\chi_{1303}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{651}\right)\) | \(e\left(\frac{277}{434}\right)\) | \(e\left(\frac{398}{651}\right)\) | \(e\left(\frac{53}{62}\right)\) | \(e\left(\frac{1229}{1302}\right)\) | \(e\left(\frac{359}{1302}\right)\) | \(e\left(\frac{199}{217}\right)\) | \(e\left(\frac{60}{217}\right)\) | \(e\left(\frac{209}{1302}\right)\) | \(e\left(\frac{39}{62}\right)\) |
| \(\chi_{1303}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{478}{651}\right)\) | \(e\left(\frac{215}{434}\right)\) | \(e\left(\frac{305}{651}\right)\) | \(e\left(\frac{53}{62}\right)\) | \(e\left(\frac{299}{1302}\right)\) | \(e\left(\frac{545}{1302}\right)\) | \(e\left(\frac{44}{217}\right)\) | \(e\left(\frac{215}{217}\right)\) | \(e\left(\frac{767}{1302}\right)\) | \(e\left(\frac{39}{62}\right)\) |
| \(\chi_{1303}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{651}\right)\) | \(e\left(\frac{215}{434}\right)\) | \(e\left(\frac{88}{651}\right)\) | \(e\left(\frac{53}{62}\right)\) | \(e\left(\frac{733}{1302}\right)\) | \(e\left(\frac{979}{1302}\right)\) | \(e\left(\frac{44}{217}\right)\) | \(e\left(\frac{215}{217}\right)\) | \(e\left(\frac{1201}{1302}\right)\) | \(e\left(\frac{39}{62}\right)\) |
| \(\chi_{1303}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{550}{651}\right)\) | \(e\left(\frac{409}{434}\right)\) | \(e\left(\frac{449}{651}\right)\) | \(e\left(\frac{27}{62}\right)\) | \(e\left(\frac{1025}{1302}\right)\) | \(e\left(\frac{845}{1302}\right)\) | \(e\left(\frac{116}{217}\right)\) | \(e\left(\frac{192}{217}\right)\) | \(e\left(\frac{365}{1302}\right)\) | \(e\left(\frac{7}{62}\right)\) |