from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(27,784))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(784\) | |
Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.28.271776353216347717810469630450516372938858574109997048774397001728.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{784}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) |
\(\chi_{784}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) |
\(\chi_{784}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) |
\(\chi_{784}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(-i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) |
\(\chi_{784}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(i\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) |
\(\chi_{784}(363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-i\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |
\(\chi_{784}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(i\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) |
\(\chi_{784}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) |
\(\chi_{784}(531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) |
\(\chi_{784}(643,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) |
\(\chi_{784}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-i\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) |
\(\chi_{784}(755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(i\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |