from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([33,22,126]))
chi.galois_orbit()
[g,chi] = znchar(Mod(87,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(132\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4235.cy | 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: | $\Q(\zeta_{132})$ |
| Fixed field: | Number field defined by a degree 132 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(-i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{89}{132}\right)\) |
| \(\chi_{8470}(593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{103}{132}\right)\) |
| \(\chi_{8470}(703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{107}{132}\right)\) |
| \(\chi_{8470}(747,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(-i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{25}{132}\right)\) |
| \(\chi_{8470}(857,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(-i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{29}{132}\right)\) |
| \(\chi_{8470}(1363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{43}{132}\right)\) |
| \(\chi_{8470}(1473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{47}{132}\right)\) |
| \(\chi_{8470}(1517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(-i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) |
| \(\chi_{8470}(1627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(-i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{101}{132}\right)\) |
| \(\chi_{8470}(2133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{115}{132}\right)\) |
| \(\chi_{8470}(2243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{119}{132}\right)\) |
| \(\chi_{8470}(2287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{37}{132}\right)\) |
| \(\chi_{8470}(2397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(-i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{41}{132}\right)\) |
| \(\chi_{8470}(3013,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) |
| \(\chi_{8470}(3057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{109}{132}\right)\) |
| \(\chi_{8470}(3167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(-i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{113}{132}\right)\) |
| \(\chi_{8470}(3673,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{127}{132}\right)\) |
| \(\chi_{8470}(3783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{131}{132}\right)\) |
| \(\chi_{8470}(3827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{49}{132}\right)\) |
| \(\chi_{8470}(3937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(-i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{53}{132}\right)\) |
| \(\chi_{8470}(4443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{67}{132}\right)\) |
| \(\chi_{8470}(4553,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{71}{132}\right)\) |
| \(\chi_{8470}(4707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{125}{132}\right)\) |
| \(\chi_{8470}(5213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{132}\right)\) |
| \(\chi_{8470}(5367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{61}{132}\right)\) |
| \(\chi_{8470}(5477,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{65}{132}\right)\) |
| \(\chi_{8470}(5983,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{79}{132}\right)\) |
| \(\chi_{8470}(6093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{83}{132}\right)\) |
| \(\chi_{8470}(6137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(-i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{1}{132}\right)\) |
| \(\chi_{8470}(6247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{132}\right)\) |
| \(\chi_{8470}(6753,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{19}{132}\right)\) |