Properties

Label 8041.134
Modulus $8041$
Conductor $8041$
Order $840$
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(840))
 
M = H._module
 
chi = DirichletCharacter(H, M([84,315,500]))
 
pari: [g,chi] = znchar(Mod(134,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(840\)
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.fw

\(\chi_{8041}(19,\cdot)\) \(\chi_{8041}(134,\cdot)\) \(\chi_{8041}(162,\cdot)\) \(\chi_{8041}(206,\cdot)\) \(\chi_{8041}(270,\cdot)\) \(\chi_{8041}(304,\cdot)\) \(\chi_{8041}(321,\cdot)\) \(\chi_{8041}(349,\cdot)\) \(\chi_{8041}(491,\cdot)\) \(\chi_{8041}(501,\cdot)\) \(\chi_{8041}(502,\cdot)\) \(\chi_{8041}(519,\cdot)\) \(\chi_{8041}(535,\cdot)\) \(\chi_{8041}(536,\cdot)\) \(\chi_{8041}(546,\cdot)\) \(\chi_{8041}(678,\cdot)\) \(\chi_{8041}(706,\cdot)\) \(\chi_{8041}(722,\cdot)\) \(\chi_{8041}(750,\cdot)\) \(\chi_{8041}(865,\cdot)\) \(\chi_{8041}(886,\cdot)\) \(\chi_{8041}(893,\cdot)\) \(\chi_{8041}(937,\cdot)\) \(\chi_{8041}(1018,\cdot)\) \(\chi_{8041}(1052,\cdot)\) \(\chi_{8041}(1062,\cdot)\) \(\chi_{8041}(1080,\cdot)\) \(\chi_{8041}(1130,\cdot)\) \(\chi_{8041}(1216,\cdot)\) \(\chi_{8041}(1250,\cdot)\) ...

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: $\Q(\zeta_{840})$
Fixed field: Number field defined by a degree 840 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3}{8}\right),e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(134, a) \) \(1\)\(1\)\(e\left(\frac{59}{140}\right)\)\(e\left(\frac{647}{840}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{131}{840}\right)\)\(e\left(\frac{23}{120}\right)\)\(e\left(\frac{79}{120}\right)\)\(e\left(\frac{37}{140}\right)\)\(e\left(\frac{227}{420}\right)\)\(e\left(\frac{97}{168}\right)\)\(e\left(\frac{103}{168}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(134,a) \;\) at \(\;a = \) e.g. 2