from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(837, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([5,72]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,837))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(837\) | |
Conductor: | \(837\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(90\) | 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: | $\Q(\zeta_{45})$ |
Fixed field: | Number field defined by a degree 90 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{837}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) |
\(\chi_{837}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) |
\(\chi_{837}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{26}{45}\right)\) |
\(\chi_{837}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) |
\(\chi_{837}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) |
\(\chi_{837}(140,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) |
\(\chi_{837}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) |
\(\chi_{837}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) |
\(\chi_{837}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) |
\(\chi_{837}(326,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) |
\(\chi_{837}(374,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) |
\(\chi_{837}(380,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) |
\(\chi_{837}(407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) |
\(\chi_{837}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) |
\(\chi_{837}(473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) |
\(\chi_{837}(500,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) |
\(\chi_{837}(560,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) |
\(\chi_{837}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) |
\(\chi_{837}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{41}{45}\right)\) |
\(\chi_{837}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) |
\(\chi_{837}(686,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) |
\(\chi_{837}(698,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{29}{45}\right)\) |
\(\chi_{837}(752,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) |
\(\chi_{837}(779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{2}{45}\right)\) |