from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4655, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([21,4,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(107,4655))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4655\) | |
| Conductor: | \(4655\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4655}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{4655}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{4655}(772,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{4655}(977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{4655}(1038,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{4655}(1437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{4655}(1642,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{4655}(1703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{4655}(1908,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{4655}(2102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{4655}(2307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{4655}(2368,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{4655}(2573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{4655}(2767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{4655}(2972,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{4655}(3033,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{4655}(3238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{4655}(3432,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{4655}(3637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{4655}(3698,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{4655}(3903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{4655}(4302,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{4655}(4363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{4655}(4568,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) |