from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([220,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,847))
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: | \(165\) | 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 165 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{847}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{46}{55}\right)\) |
| \(\chi_{847}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{847}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{49}{55}\right)\) |
| \(\chi_{847}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{55}\right)\) |
| \(\chi_{847}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{18}{55}\right)\) |
| \(\chi_{847}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{29}{55}\right)\) |
| \(\chi_{847}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{32}{55}\right)\) |
| \(\chi_{847}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{48}{55}\right)\) |
| \(\chi_{847}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{847}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{847}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{847}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{19}{55}\right)\) |
| \(\chi_{847}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{847}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{31}{55}\right)\) |
| \(\chi_{847}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{847}(170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{847}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{55}\right)\) |
| \(\chi_{847}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{16}{55}\right)\) |
| \(\chi_{847}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{48}{55}\right)\) |
| \(\chi_{847}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{847}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{847}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{847}(240,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{847}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{52}{55}\right)\) |
| \(\chi_{847}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{19}{55}\right)\) |
| \(\chi_{847}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{36}{55}\right)\) |
| \(\chi_{847}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{847}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{54}{55}\right)\) |
| \(\chi_{847}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{847}(312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{16}{55}\right)\) |
| \(\chi_{847}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{38}{55}\right)\) |