Properties

Label 4284.1339
Modulus $4284$
Conductor $4284$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4284, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,8,4,3]))
 
pari: [g,chi] = znchar(Mod(1339,4284))
 

Basic properties

Modulus: \(4284\)
Conductor: \(4284\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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)
 

Galois orbit 4284.dz

\(\chi_{4284}(319,\cdot)\) \(\chi_{4284}(1075,\cdot)\) \(\chi_{4284}(1339,\cdot)\) \(\chi_{4284}(2095,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.120538181723674419070377725952.2

Values on generators

\((2143,3809,1837,1261)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4284 }(1339, a) \) \(-1\)\(1\)\(i\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4284 }(1339,a) \;\) at \(\;a = \) e.g. 2