from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([55,147]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,847))
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: | even | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{847}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{27}{55}\right)\) |
| \(\chi_{847}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{847}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{847}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{4}{55}\right)\) |
| \(\chi_{847}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{32}{55}\right)\) |
| \(\chi_{847}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{18}{55}\right)\) |
| \(\chi_{847}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{26}{55}\right)\) |
| \(\chi_{847}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{49}{55}\right)\) |
| \(\chi_{847}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{847}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{46}{55}\right)\) |
| \(\chi_{847}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{14}{55}\right)\) |
| \(\chi_{847}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{27}{55}\right)\) |
| \(\chi_{847}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{847}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{847}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{17}{55}\right)\) |
| \(\chi_{847}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{4}{55}\right)\) |
| \(\chi_{847}(178,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{28}{55}\right)\) |
| \(\chi_{847}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{26}{55}\right)\) |
| \(\chi_{847}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{24}{55}\right)\) |
| \(\chi_{847}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{847}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{41}{55}\right)\) |
| \(\chi_{847}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{12}{55}\right)\) |
| \(\chi_{847}(250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{14}{55}\right)\) |
| \(\chi_{847}(255,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{13}{55}\right)\) |
| \(\chi_{847}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{847}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{34}{55}\right)\) |
| \(\chi_{847}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{17}{55}\right)\) |
| \(\chi_{847}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{28}{55}\right)\) |
| \(\chi_{847}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{21}{55}\right)\) |
| \(\chi_{847}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{7}{55}\right)\) |
| \(\chi_{847}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{24}{55}\right)\) |