from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,784))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(784\) | |
Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 392.z | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\Q(\zeta_{21})\) |
Fixed field: | 42.42.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{784}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) |
\(\chi_{784}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) |
\(\chi_{784}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) |
\(\chi_{784}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) |
\(\chi_{784}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) |
\(\chi_{784}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) |
\(\chi_{784}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) |
\(\chi_{784}(457,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) |
\(\chi_{784}(473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) |
\(\chi_{784}(585,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) |
\(\chi_{784}(681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) |
\(\chi_{784}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) |