Properties

Label 8041.581
Modulus $8041$
Conductor $8041$
Order $560$
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(560))
 
M = H._module
 
chi = DirichletCharacter(H, M([336,35,200]))
 
pari: [g,chi] = znchar(Mod(581,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(560\)
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.ft

\(\chi_{8041}(27,\cdot)\) \(\chi_{8041}(75,\cdot)\) \(\chi_{8041}(82,\cdot)\) \(\chi_{8041}(108,\cdot)\) \(\chi_{8041}(113,\cdot)\) \(\chi_{8041}(125,\cdot)\) \(\chi_{8041}(180,\cdot)\) \(\chi_{8041}(328,\cdot)\) \(\chi_{8041}(333,\cdot)\) \(\chi_{8041}(346,\cdot)\) \(\chi_{8041}(432,\cdot)\) \(\chi_{8041}(500,\cdot)\) \(\chi_{8041}(555,\cdot)\) \(\chi_{8041}(581,\cdot)\) \(\chi_{8041}(598,\cdot)\) \(\chi_{8041}(641,\cdot)\) \(\chi_{8041}(653,\cdot)\) \(\chi_{8041}(720,\cdot)\) \(\chi_{8041}(753,\cdot)\) \(\chi_{8041}(796,\cdot)\) \(\chi_{8041}(806,\cdot)\) \(\chi_{8041}(819,\cdot)\) \(\chi_{8041}(839,\cdot)\) \(\chi_{8041}(856,\cdot)\) \(\chi_{8041}(862,\cdot)\) \(\chi_{8041}(911,\cdot)\) \(\chi_{8041}(1059,\cdot)\) \(\chi_{8041}(1083,\cdot)\) \(\chi_{8041}(1193,\cdot)\) \(\chi_{8041}(1236,\cdot)\) ...

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: $\Q(\zeta_{560})$
Fixed field: Number field defined by a degree 560 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{16}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(581, a) \) \(1\)\(1\)\(e\left(\frac{33}{280}\right)\)\(e\left(\frac{123}{560}\right)\)\(e\left(\frac{33}{140}\right)\)\(e\left(\frac{359}{560}\right)\)\(e\left(\frac{27}{80}\right)\)\(e\left(\frac{31}{80}\right)\)\(e\left(\frac{99}{280}\right)\)\(e\left(\frac{123}{280}\right)\)\(e\left(\frac{85}{112}\right)\)\(e\left(\frac{51}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(581,a) \;\) at \(\;a = \) e.g. 2