Properties

Label 4024.267
Modulus $4024$
Conductor $4024$
Order $502$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, base_ring=CyclotomicField(502))
 
M = H._module
 
chi = DirichletCharacter(H, M([251,251,331]))
 
pari: [g,chi] = znchar(Mod(267,4024))
 

Basic properties

Modulus: \(4024\)
Conductor: \(4024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(502\)
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 4024.j

\(\chi_{4024}(19,\cdot)\) \(\chi_{4024}(35,\cdot)\) \(\chi_{4024}(51,\cdot)\) \(\chi_{4024}(107,\cdot)\) \(\chi_{4024}(115,\cdot)\) \(\chi_{4024}(123,\cdot)\) \(\chi_{4024}(139,\cdot)\) \(\chi_{4024}(163,\cdot)\) \(\chi_{4024}(171,\cdot)\) \(\chi_{4024}(179,\cdot)\) \(\chi_{4024}(187,\cdot)\) \(\chi_{4024}(195,\cdot)\) \(\chi_{4024}(203,\cdot)\) \(\chi_{4024}(211,\cdot)\) \(\chi_{4024}(227,\cdot)\) \(\chi_{4024}(235,\cdot)\) \(\chi_{4024}(251,\cdot)\) \(\chi_{4024}(259,\cdot)\) \(\chi_{4024}(267,\cdot)\) \(\chi_{4024}(307,\cdot)\) \(\chi_{4024}(315,\cdot)\) \(\chi_{4024}(331,\cdot)\) \(\chi_{4024}(347,\cdot)\) \(\chi_{4024}(371,\cdot)\) \(\chi_{4024}(395,\cdot)\) \(\chi_{4024}(403,\cdot)\) \(\chi_{4024}(411,\cdot)\) \(\chi_{4024}(419,\cdot)\) \(\chi_{4024}(451,\cdot)\) \(\chi_{4024}(459,\cdot)\) ...

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: $\Q(\zeta_{251})$
Fixed field: Number field defined by a degree 502 polynomial (not computed)

Values on generators

\((1007,2013,2017)\) → \((-1,-1,e\left(\frac{331}{502}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4024 }(267, a) \) \(1\)\(1\)\(e\left(\frac{216}{251}\right)\)\(e\left(\frac{40}{251}\right)\)\(e\left(\frac{103}{502}\right)\)\(e\left(\frac{181}{251}\right)\)\(e\left(\frac{174}{251}\right)\)\(e\left(\frac{131}{502}\right)\)\(e\left(\frac{5}{251}\right)\)\(e\left(\frac{499}{502}\right)\)\(e\left(\frac{135}{502}\right)\)\(e\left(\frac{33}{502}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4024 }(267,a) \;\) at \(\;a = \) e.g. 2