from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,83]))
chi.galois_orbit()
[g,chi] = znchar(Mod(324,6001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6001\) | |
| Conductor: | \(353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(88\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 353.j | 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: | $\Q(\zeta_{88})$ |
| Fixed field: | Number field defined by a degree 88 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6001}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{6001}(426,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{6001}(698,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{6001}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{6001}(834,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{6001}(1038,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{6001}(1140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{6001}(1344,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{6001}(1395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{6001}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{6001}(1480,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{6001}(1684,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{6001}(1786,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{6001}(1990,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{6001}(2007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{6001}(2126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{6001}(2398,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{6001}(2500,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{6001}(2908,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{6001}(3469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{6001}(3486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{6001}(3639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{6001}(3707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{6001}(3724,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{6001}(3792,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{6001}(3894,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{6001}(4234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{6001}(4268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{6001}(4319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{6001}(4506,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{6001}(4557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{22}\right)\) |