from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([110,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,847))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(847\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{847}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{101}{110}\right)\) |
| \(\chi_{847}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{57}{110}\right)\) |
| \(\chi_{847}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{73}{110}\right)\) |
| \(\chi_{847}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{59}{110}\right)\) |
| \(\chi_{847}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{21}{110}\right)\) |
| \(\chi_{847}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{87}{110}\right)\) |
| \(\chi_{847}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{39}{110}\right)\) |
| \(\chi_{847}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{53}{110}\right)\) |
| \(\chi_{847}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{71}{110}\right)\) |
| \(\chi_{847}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{17}{110}\right)\) |
| \(\chi_{847}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{93}{110}\right)\) |
| \(\chi_{847}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{49}{110}\right)\) |
| \(\chi_{847}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{101}{110}\right)\) |
| \(\chi_{847}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{47}{110}\right)\) |
| \(\chi_{847}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{29}{110}\right)\) |
| \(\chi_{847}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{73}{110}\right)\) |
| \(\chi_{847}(156,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{41}{110}\right)\) |
| \(\chi_{847}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{87}{110}\right)\) |
| \(\chi_{847}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{110}\right)\) |
| \(\chi_{847}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{39}{110}\right)\) |
| \(\chi_{847}(200,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{71}{110}\right)\) |
| \(\chi_{847}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{110}\right)\) |
| \(\chi_{847}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{19}{110}\right)\) |
| \(\chi_{847}(228,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{93}{110}\right)\) |
| \(\chi_{847}(249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{110}\right)\) |
| \(\chi_{847}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{23}{110}\right)\) |
| \(\chi_{847}(270,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{29}{110}\right)\) |
| \(\chi_{847}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{41}{110}\right)\) |
| \(\chi_{847}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{9}{110}\right)\) |
| \(\chi_{847}(305,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{3}{110}\right)\) |
| \(\chi_{847}(310,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{91}{110}\right)\) |