Properties

Label 8041.5147
Modulus $8041$
Conductor $8041$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,21,22]))
 
pari: [g,chi] = znchar(Mod(5147,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.ec

\(\chi_{8041}(98,\cdot)\) \(\chi_{8041}(846,\cdot)\) \(\chi_{8041}(1517,\cdot)\) \(\chi_{8041}(1594,\cdot)\) \(\chi_{8041}(1781,\cdot)\) \(\chi_{8041}(1968,\cdot)\) \(\chi_{8041}(2155,\cdot)\) \(\chi_{8041}(2265,\cdot)\) \(\chi_{8041}(2342,\cdot)\) \(\chi_{8041}(3013,\cdot)\) \(\chi_{8041}(3200,\cdot)\) \(\chi_{8041}(3387,\cdot)\) \(\chi_{8041}(3574,\cdot)\) \(\chi_{8041}(3761,\cdot)\) \(\chi_{8041}(4025,\cdot)\) \(\chi_{8041}(4586,\cdot)\) \(\chi_{8041}(5147,\cdot)\) \(\chi_{8041}(5444,\cdot)\) \(\chi_{8041}(6005,\cdot)\) \(\chi_{8041}(6082,\cdot)\) \(\chi_{8041}(6269,\cdot)\) \(\chi_{8041}(6566,\cdot)\) \(\chi_{8041}(7501,\cdot)\) \(\chi_{8041}(7688,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((6580,2366,562)\) → \((-1,i,e\left(\frac{11}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(5147, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{55}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(5147,a) \;\) at \(\;a = \) e.g. 2