Properties

Label 8026.9
Modulus $8026$
Conductor $4013$
Order $2006$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, base_ring=CyclotomicField(2006))
 
M = H._module
 
chi = DirichletCharacter(H, M([1745]))
 
pari: [g,chi] = znchar(Mod(9,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2006\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.k

\(\chi_{8026}(9,\cdot)\) \(\chi_{8026}(13,\cdot)\) \(\chi_{8026}(15,\cdot)\) \(\chi_{8026}(17,\cdot)\) \(\chi_{8026}(25,\cdot)\) \(\chi_{8026}(31,\cdot)\) \(\chi_{8026}(47,\cdot)\) \(\chi_{8026}(59,\cdot)\) \(\chi_{8026}(63,\cdot)\) \(\chi_{8026}(69,\cdot)\) \(\chi_{8026}(71,\cdot)\) \(\chi_{8026}(89,\cdot)\) \(\chi_{8026}(91,\cdot)\) \(\chi_{8026}(97,\cdot)\) \(\chi_{8026}(99,\cdot)\) \(\chi_{8026}(105,\cdot)\) \(\chi_{8026}(115,\cdot)\) \(\chi_{8026}(127,\cdot)\) \(\chi_{8026}(143,\cdot)\) \(\chi_{8026}(165,\cdot)\) \(\chi_{8026}(171,\cdot)\) \(\chi_{8026}(175,\cdot)\) \(\chi_{8026}(187,\cdot)\) \(\chi_{8026}(191,\cdot)\) \(\chi_{8026}(217,\cdot)\) \(\chi_{8026}(237,\cdot)\) \(\chi_{8026}(247,\cdot)\) \(\chi_{8026}(257,\cdot)\) \(\chi_{8026}(271,\cdot)\) \(\chi_{8026}(275,\cdot)\) ...

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: $\Q(\zeta_{1003})$
Fixed field: Number field defined by a degree 2006 polynomial (not computed)

Values on generators

\(4015\) → \(e\left(\frac{1745}{2006}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(9, a) \) \(1\)\(1\)\(e\left(\frac{1923}{2006}\right)\)\(e\left(\frac{851}{2006}\right)\)\(e\left(\frac{154}{1003}\right)\)\(e\left(\frac{920}{1003}\right)\)\(e\left(\frac{403}{1003}\right)\)\(e\left(\frac{624}{1003}\right)\)\(e\left(\frac{384}{1003}\right)\)\(e\left(\frac{169}{1003}\right)\)\(e\left(\frac{291}{1003}\right)\)\(e\left(\frac{225}{2006}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(9,a) \;\) at \(\;a = \) e.g. 2