from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(163, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,163))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(163\) | |
| Conductor: | \(163\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(162\) | 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: | $\Q(\zeta_{81})$ |
| Fixed field: | Number field defined by a degree 162 polynomial (not computed) |
First 31 of 54 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{163}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{162}\right)\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{47}{162}\right)\) |
| \(\chi_{163}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{83}{162}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{49}{162}\right)\) |
| \(\chi_{163}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{83}{162}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{29}{162}\right)\) |
| \(\chi_{163}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{52}{81}\right)\) | \(e\left(\frac{103}{162}\right)\) |
| \(\chi_{163}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{162}\right)\) | \(e\left(\frac{35}{162}\right)\) | \(e\left(\frac{22}{81}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{67}{162}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{143}{162}\right)\) |
| \(\chi_{163}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{145}{162}\right)\) |
| \(\chi_{163}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{151}{162}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{43}{162}\right)\) |
| \(\chi_{163}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{97}{162}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{16}{81}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{151}{162}\right)\) |
| \(\chi_{163}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{26}{81}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{35}{162}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{7}{162}\right)\) |
| \(\chi_{163}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{162}\right)\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{40}{81}\right)\) | \(e\left(\frac{73}{162}\right)\) |
| \(\chi_{163}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{162}\right)\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{139}{162}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{17}{81}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{125}{162}\right)\) |
| \(\chi_{163}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{13}{162}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{35}{162}\right)\) |
| \(\chi_{163}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{162}\right)\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{55}{81}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{127}{162}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{47}{81}\right)\) | \(e\left(\frac{35}{81}\right)\) | \(e\left(\frac{155}{162}\right)\) |
| \(\chi_{163}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{162}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{161}{162}\right)\) |
| \(\chi_{163}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{7}{162}\right)\) | \(e\left(\frac{53}{81}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{143}{162}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{61}{162}\right)\) |
| \(\chi_{163}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{149}{162}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{127}{162}\right)\) |
| \(\chi_{163}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{162}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{23}{162}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{64}{81}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{37}{162}\right)\) |
| \(\chi_{163}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{162}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{43}{162}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{38}{81}\right)\) | \(e\left(\frac{41}{162}\right)\) |
| \(\chi_{163}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{162}\right)\) | \(e\left(\frac{127}{162}\right)\) | \(e\left(\frac{59}{81}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{19}{162}\right)\) |
| \(\chi_{163}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{79}{162}\right)\) | \(e\left(\frac{8}{81}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{79}{81}\right)\) | \(e\left(\frac{64}{81}\right)\) | \(e\left(\frac{133}{162}\right)\) |
| \(\chi_{163}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{162}\right)\) | \(e\left(\frac{131}{162}\right)\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{61}{162}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{77}{162}\right)\) |
| \(\chi_{163}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{162}\right)\) | \(e\left(\frac{67}{162}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{119}{162}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{67}{81}\right)\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{121}{162}\right)\) |
| \(\chi_{163}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{162}\right)\) | \(e\left(\frac{109}{162}\right)\) | \(e\left(\frac{50}{81}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{5}{162}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{28}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{1}{162}\right)\) |
| \(\chi_{163}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{162}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{46}{81}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{37}{162}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{137}{162}\right)\) |
| \(\chi_{163}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{37}{162}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{131}{162}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{91}{162}\right)\) |
| \(\chi_{163}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{137}{162}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{83}{162}\right)\) |
| \(\chi_{163}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{162}\right)\) | \(e\left(\frac{61}{162}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{73}{81}\right)\) | \(e\left(\frac{115}{162}\right)\) |
| \(\chi_{163}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{34}{81}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{59}{162}\right)\) |
| \(\chi_{163}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{162}\right)\) | \(e\left(\frac{139}{162}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{155}{162}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{31}{162}\right)\) |
| \(\chi_{163}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{13}{162}\right)\) | \(e\left(\frac{29}{81}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{11}{162}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{67}{162}\right)\) |
| \(\chi_{163}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{162}\right)\) | \(e\left(\frac{23}{162}\right)\) | \(e\left(\frac{70}{81}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{162}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{23}{81}\right)\) | \(e\left(\frac{74}{81}\right)\) | \(e\left(\frac{131}{162}\right)\) |