Properties

Label 8041.2704
Modulus $8041$
Conductor $473$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([126,0,20]))
 
pari: [g,chi] = znchar(Mod(2704,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(339,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.ee

\(\chi_{8041}(103,\cdot)\) \(\chi_{8041}(273,\cdot)\) \(\chi_{8041}(324,\cdot)\) \(\chi_{8041}(443,\cdot)\) \(\chi_{8041}(511,\cdot)\) \(\chi_{8041}(698,\cdot)\) \(\chi_{8041}(834,\cdot)\) \(\chi_{8041}(885,\cdot)\) \(\chi_{8041}(1004,\cdot)\) \(\chi_{8041}(1072,\cdot)\) \(\chi_{8041}(1565,\cdot)\) \(\chi_{8041}(1786,\cdot)\) \(\chi_{8041}(1820,\cdot)\) \(\chi_{8041}(1973,\cdot)\) \(\chi_{8041}(2160,\cdot)\) \(\chi_{8041}(2347,\cdot)\) \(\chi_{8041}(2517,\cdot)\) \(\chi_{8041}(2534,\cdot)\) \(\chi_{8041}(2568,\cdot)\) \(\chi_{8041}(2704,\cdot)\) \(\chi_{8041}(2891,\cdot)\) \(\chi_{8041}(3078,\cdot)\) \(\chi_{8041}(3248,\cdot)\) \(\chi_{8041}(3265,\cdot)\) \(\chi_{8041}(3282,\cdot)\) \(\chi_{8041}(3435,\cdot)\) \(\chi_{8041}(3622,\cdot)\) \(\chi_{8041}(3809,\cdot)\) \(\chi_{8041}(3996,\cdot)\) \(\chi_{8041}(4013,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{3}{5}\right),1,e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(2704, a) \) \(1\)\(1\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{94}{105}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{83}{105}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(2704,a) \;\) at \(\;a = \) e.g. 2