from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,53]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,1340))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1340\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 67.h | 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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1340}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) |
\(\chi_{1340}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
\(\chi_{1340}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |
\(\chi_{1340}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
\(\chi_{1340}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |
\(\chi_{1340}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
\(\chi_{1340}(381,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
\(\chi_{1340}(481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
\(\chi_{1340}(501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
\(\chi_{1340}(621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
\(\chi_{1340}(681,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
\(\chi_{1340}(701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
\(\chi_{1340}(721,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
\(\chi_{1340}(781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |
\(\chi_{1340}(861,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
\(\chi_{1340}(921,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) |
\(\chi_{1340}(1001,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
\(\chi_{1340}(1141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
\(\chi_{1340}(1301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
\(\chi_{1340}(1321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |