from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([28,63,4]))
chi.galois_orbit()
[g,chi] = znchar(Mod(58,2205))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2205\) | |
Conductor: | \(2205\) | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2205}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) |
\(\chi_{2205}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) |
\(\chi_{2205}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) |
\(\chi_{2205}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) |
\(\chi_{2205}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
\(\chi_{2205}(562,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) |
\(\chi_{2205}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) |
\(\chi_{2205}(688,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |
\(\chi_{2205}(718,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
\(\chi_{2205}(877,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
\(\chi_{2205}(907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
\(\chi_{2205}(1003,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) |
\(\chi_{2205}(1033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) |
\(\chi_{2205}(1192,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) |
\(\chi_{2205}(1222,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) |
\(\chi_{2205}(1318,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) |
\(\chi_{2205}(1348,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) |
\(\chi_{2205}(1507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
\(\chi_{2205}(1633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) |
\(\chi_{2205}(1663,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) |
\(\chi_{2205}(1822,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
\(\chi_{2205}(1852,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) |
\(\chi_{2205}(1948,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
\(\chi_{2205}(2167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |