from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,14,21,68]))
chi.galois_orbit()
[g,chi] = znchar(Mod(137,8820))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(2205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(84\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2205.en | 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 | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8820}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{8820}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{8820}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{8820}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{8820}(1913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{8820}(2153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{8820}(2417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{8820}(2657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{8820}(3173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{8820}(3413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{8820}(3677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{8820}(3917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{8820}(4433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{8820}(4937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{8820}(5177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{8820}(5693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{8820}(5933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{8820}(6197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{8820}(6953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{8820}(7193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{8820}(7457,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{8820}(7697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{8820}(8453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{8820}(8717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) |