Properties

Label 3479.449
Modulus $3479$
Conductor $3479$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3479, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,3]))
 
pari: [g,chi] = znchar(Mod(449,3479))
 

Basic properties

Modulus: \(3479\)
Conductor: \(3479\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
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 3479.ba

\(\chi_{3479}(449,\cdot)\) \(\chi_{3479}(477,\cdot)\) \(\chi_{3479}(1170,\cdot)\) \(\chi_{3479}(2227,\cdot)\) \(\chi_{3479}(2808,\cdot)\) \(\chi_{3479}(3165,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((640,1569)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 3479 }(449, a) \) \(-1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3479 }(449,a) \;\) at \(\;a = \) e.g. 2