Properties

Label 784.bj
Modulus $784$
Conductor $784$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,7,4]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(43,784))
 
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{784}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(i\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{784}(155,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(i\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{784}(211,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(-i\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{784}(267,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(i\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{784}(323,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(-i\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{784}(379,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(i\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{784}(435,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(-i\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{784}(547,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(-i\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{784}(603,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(i\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{784}(659,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(-i\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{784}(715,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(i\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{784}(771,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(-i\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\)