from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7056, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,21,14,80]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,7056))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(7056\) | |
Conductor: | \(7056\) | 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 | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{7056}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(-1\) | \(e\left(\frac{61}{84}\right)\) |
\(\chi_{7056}(515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(-1\) | \(e\left(\frac{79}{84}\right)\) |
\(\chi_{7056}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(-1\) | \(e\left(\frac{29}{84}\right)\) |
\(\chi_{7056}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(-1\) | \(e\left(\frac{13}{84}\right)\) |
\(\chi_{7056}(1283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(-1\) | \(e\left(\frac{23}{84}\right)\) |
\(\chi_{7056}(1523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(-1\) | \(e\left(\frac{31}{84}\right)\) |
\(\chi_{7056}(1787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(-1\) | \(e\left(\frac{17}{84}\right)\) |
\(\chi_{7056}(2291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(-1\) | \(e\left(\frac{11}{84}\right)\) |
\(\chi_{7056}(2531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(-1\) | \(e\left(\frac{67}{84}\right)\) |
\(\chi_{7056}(2795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(-1\) | \(e\left(\frac{5}{84}\right)\) |
\(\chi_{7056}(3035,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(-1\) | \(e\left(\frac{1}{84}\right)\) |
\(\chi_{7056}(3299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(-1\) | \(e\left(\frac{83}{84}\right)\) |
\(\chi_{7056}(3539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(-1\) | \(e\left(\frac{19}{84}\right)\) |
\(\chi_{7056}(4043,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(-1\) | \(e\left(\frac{37}{84}\right)\) |
\(\chi_{7056}(4307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(-1\) | \(e\left(\frac{71}{84}\right)\) |
\(\chi_{7056}(4547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(-1\) | \(e\left(\frac{55}{84}\right)\) |
\(\chi_{7056}(4811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(-1\) | \(e\left(\frac{65}{84}\right)\) |
\(\chi_{7056}(5051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(-1\) | \(e\left(\frac{73}{84}\right)\) |
\(\chi_{7056}(5315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(-1\) | \(e\left(\frac{59}{84}\right)\) |
\(\chi_{7056}(5819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(-1\) | \(e\left(\frac{53}{84}\right)\) |
\(\chi_{7056}(6059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(-1\) | \(e\left(\frac{25}{84}\right)\) |
\(\chi_{7056}(6323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(-1\) | \(e\left(\frac{47}{84}\right)\) |
\(\chi_{7056}(6563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(-1\) | \(e\left(\frac{43}{84}\right)\) |
\(\chi_{7056}(6827,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(-1\) | \(e\left(\frac{41}{84}\right)\) |