from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3963, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([0,113]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,3963))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3963\) | |
| Conductor: | \(1321\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(264\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1321.bb | 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: | $\Q(\zeta_{264})$ |
| Fixed field: | Number field defined by a degree 264 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3963}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{221}{264}\right)\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{157}{264}\right)\) | \(-i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{97}{264}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{127}{264}\right)\) | \(i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{139}{264}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{203}{264}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{35}{264}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(130,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(-i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{5}{264}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{223}{264}\right)\) | \(i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{163}{264}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{247}{264}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{235}{264}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(385,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{251}{264}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{71}{264}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{151}{264}\right)\) | \(i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{23}{264}\right)\) | \(i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(556,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{133}{264}\right)\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(574,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{109}{264}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(904,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(-i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(913,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{245}{264}\right)\) | \(-i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(955,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{37}{264}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1006,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{79}{264}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1054,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{61}{264}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{233}{264}\right)\) | \(-i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{65}{264}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1282,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1360,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1390,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{197}{264}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1435,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{101}{264}\right)\) | \(-i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1489,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{59}{264}\right)\) | \(i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{23}{264}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1588,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1636,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{127}{264}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{181}{264}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3963}(1729,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(-i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{53}{264}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3963}(1738,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{257}{264}\right)\) | \(-i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{221}{264}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{1}{3}\right)\) |