from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1421, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([52,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1421))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1421\) | |
Conductor: | \(1421\) | 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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1421}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) |
\(\chi_{1421}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) |
\(\chi_{1421}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) |
\(\chi_{1421}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) |
\(\chi_{1421}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) |
\(\chi_{1421}(240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) |
\(\chi_{1421}(340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) |
\(\chi_{1421}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) |
\(\chi_{1421}(380,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) |
\(\chi_{1421}(396,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
\(\chi_{1421}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
\(\chi_{1421}(450,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |
\(\chi_{1421}(485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) |
\(\chi_{1421}(627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
\(\chi_{1421}(711,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) |
\(\chi_{1421}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) |
\(\chi_{1421}(1012,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) |
\(\chi_{1421}(1087,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
\(\chi_{1421}(1129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
\(\chi_{1421}(1150,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
\(\chi_{1421}(1187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
\(\chi_{1421}(1199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) |
\(\chi_{1421}(1348,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |
\(\chi_{1421}(1360,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) |