Properties

Label 8041.626
Modulus $8041$
Conductor $8041$
Order $336$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,189,320]))
 
pari: [g,chi] = znchar(Mod(626,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
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)
 

Galois orbit 8041.fl

\(\chi_{8041}(10,\cdot)\) \(\chi_{8041}(109,\cdot)\) \(\chi_{8041}(142,\cdot)\) \(\chi_{8041}(197,\cdot)\) \(\chi_{8041}(296,\cdot)\) \(\chi_{8041}(318,\cdot)\) \(\chi_{8041}(384,\cdot)\) \(\chi_{8041}(439,\cdot)\) \(\chi_{8041}(483,\cdot)\) \(\chi_{8041}(615,\cdot)\) \(\chi_{8041}(626,\cdot)\) \(\chi_{8041}(670,\cdot)\) \(\chi_{8041}(703,\cdot)\) \(\chi_{8041}(857,\cdot)\) \(\chi_{8041}(912,\cdot)\) \(\chi_{8041}(1099,\cdot)\) \(\chi_{8041}(1132,\cdot)\) \(\chi_{8041}(1176,\cdot)\) \(\chi_{8041}(1264,\cdot)\) \(\chi_{8041}(1561,\cdot)\) \(\chi_{8041}(1605,\cdot)\) \(\chi_{8041}(1737,\cdot)\) \(\chi_{8041}(1858,\cdot)\) \(\chi_{8041}(1880,\cdot)\) \(\chi_{8041}(2001,\cdot)\) \(\chi_{8041}(2034,\cdot)\) \(\chi_{8041}(2045,\cdot)\) \(\chi_{8041}(2122,\cdot)\) \(\chi_{8041}(2188,\cdot)\) \(\chi_{8041}(2353,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(626, a) \) \(1\)\(1\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{173}{336}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{209}{336}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{15}{56}\right)\)\(e\left(\frac{5}{168}\right)\)\(e\left(\frac{239}{336}\right)\)\(e\left(\frac{233}{336}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(626,a) \;\) at \(\;a = \) e.g. 2