Properties

Label 8041.3409
Modulus $8041$
Conductor $8041$
Order $168$
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(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([84,21,52]))
 
pari: [g,chi] = znchar(Mod(3409,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
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.eu

\(\chi_{8041}(76,\cdot)\) \(\chi_{8041}(263,\cdot)\) \(\chi_{8041}(406,\cdot)\) \(\chi_{8041}(417,\cdot)\) \(\chi_{8041}(450,\cdot)\) \(\chi_{8041}(593,\cdot)\) \(\chi_{8041}(648,\cdot)\) \(\chi_{8041}(835,\cdot)\) \(\chi_{8041}(1022,\cdot)\) \(\chi_{8041}(1209,\cdot)\) \(\chi_{8041}(1352,\cdot)\) \(\chi_{8041}(1396,\cdot)\) \(\chi_{8041}(1539,\cdot)\) \(\chi_{8041}(2133,\cdot)\) \(\chi_{8041}(2463,\cdot)\) \(\chi_{8041}(2694,\cdot)\) \(\chi_{8041}(3079,\cdot)\) \(\chi_{8041}(3211,\cdot)\) \(\chi_{8041}(3255,\cdot)\) \(\chi_{8041}(3409,\cdot)\) \(\chi_{8041}(3640,\cdot)\) \(\chi_{8041}(3959,\cdot)\) \(\chi_{8041}(4146,\cdot)\) \(\chi_{8041}(4157,\cdot)\) \(\chi_{8041}(4190,\cdot)\) \(\chi_{8041}(4201,\cdot)\) \(\chi_{8041}(4333,\cdot)\) \(\chi_{8041}(4377,\cdot)\) \(\chi_{8041}(4520,\cdot)\) \(\chi_{8041}(4707,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(3409, a) \) \(1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{73}{168}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{61}{168}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{163}{168}\right)\)\(e\left(\frac{109}{168}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(3409,a) \;\) at \(\;a = \) e.g. 2