Properties

Label 8041.2314
Modulus $8041$
Conductor $8041$
Order $280$
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(280))
 
M = H._module
 
chi = DirichletCharacter(H, M([56,245,120]))
 
pari: [g,chi] = znchar(Mod(2314,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
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.fh

\(\chi_{8041}(59,\cdot)\) \(\chi_{8041}(213,\cdot)\) \(\chi_{8041}(236,\cdot)\) \(\chi_{8041}(355,\cdot)\) \(\chi_{8041}(434,\cdot)\) \(\chi_{8041}(570,\cdot)\) \(\chi_{8041}(729,\cdot)\) \(\chi_{8041}(790,\cdot)\) \(\chi_{8041}(852,\cdot)\) \(\chi_{8041}(944,\cdot)\) \(\chi_{8041}(950,\cdot)\) \(\chi_{8041}(1005,\cdot)\) \(\chi_{8041}(1182,\cdot)\) \(\chi_{8041}(1215,\cdot)\) \(\chi_{8041}(1301,\cdot)\) \(\chi_{8041}(1368,\cdot)\) \(\chi_{8041}(1521,\cdot)\) \(\chi_{8041}(1589,\cdot)\) \(\chi_{8041}(1675,\cdot)\) \(\chi_{8041}(1681,\cdot)\) \(\chi_{8041}(1698,\cdot)\) \(\chi_{8041}(1736,\cdot)\) \(\chi_{8041}(1896,\cdot)\) \(\chi_{8041}(2099,\cdot)\) \(\chi_{8041}(2161,\cdot)\) \(\chi_{8041}(2314,\cdot)\) \(\chi_{8041}(2412,\cdot)\) \(\chi_{8041}(2429,\cdot)\) \(\chi_{8041}(2467,\cdot)\) \(\chi_{8041}(2535,\cdot)\) ...

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: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{7}{8}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(2314, a) \) \(1\)\(1\)\(e\left(\frac{3}{140}\right)\)\(e\left(\frac{253}{280}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{249}{280}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{9}{140}\right)\)\(e\left(\frac{113}{140}\right)\)\(e\left(\frac{51}{56}\right)\)\(e\left(\frac{53}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(2314,a) \;\) at \(\;a = \) e.g. 2