from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([4,63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,2009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(2009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(84\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{84})$ |
| Fixed field: | Number field defined by a degree 84 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2009}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{2009}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{2009}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{2009}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{2009}(319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{2009}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{2009}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{2009}(583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{2009}(688,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{2009}(788,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{2009}(870,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{2009}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{2009}(975,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{2009}(1075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{2009}(1180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{2009}(1262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{2009}(1362,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{2009}(1444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{2009}(1467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{2009}(1649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{2009}(1731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{2009}(1754,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{2009}(1836,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{2009}(1936,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |