from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,89))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(89\) | |
| Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(88\) | 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: | $\Q(\zeta_{88})$ |
| Fixed field: | Number field defined by a degree 88 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{89}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{89}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{89}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{89}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{89}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{89}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{89}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{89}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{89}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{89}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{89}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{89}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{89}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{89}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{89}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{89}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{89}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{89}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{89}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{89}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{89}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{89}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{89}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{89}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{89}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{89}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{89}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{89}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{89}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{89}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{89}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |