Properties

Label 8026.7
Modulus $8026$
Conductor $4013$
Order $1003$
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([1306]))
 
pari: [g,chi] = znchar(Mod(7,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1003\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.j

\(\chi_{8026}(7,\cdot)\) \(\chi_{8026}(11,\cdot)\) \(\chi_{8026}(19,\cdot)\) \(\chi_{8026}(41,\cdot)\) \(\chi_{8026}(43,\cdot)\) \(\chi_{8026}(49,\cdot)\) \(\chi_{8026}(61,\cdot)\) \(\chi_{8026}(67,\cdot)\) \(\chi_{8026}(73,\cdot)\) \(\chi_{8026}(77,\cdot)\) \(\chi_{8026}(81,\cdot)\) \(\chi_{8026}(83,\cdot)\) \(\chi_{8026}(87,\cdot)\) \(\chi_{8026}(109,\cdot)\) \(\chi_{8026}(111,\cdot)\) \(\chi_{8026}(117,\cdot)\) \(\chi_{8026}(121,\cdot)\) \(\chi_{8026}(133,\cdot)\) \(\chi_{8026}(135,\cdot)\) \(\chi_{8026}(139,\cdot)\) \(\chi_{8026}(145,\cdot)\) \(\chi_{8026}(153,\cdot)\) \(\chi_{8026}(169,\cdot)\) \(\chi_{8026}(173,\cdot)\) \(\chi_{8026}(179,\cdot)\) \(\chi_{8026}(185,\cdot)\) \(\chi_{8026}(195,\cdot)\) \(\chi_{8026}(199,\cdot)\) \(\chi_{8026}(209,\cdot)\) \(\chi_{8026}(211,\cdot)\) ...

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 1003 polynomial (not computed)

Values on generators

\(4015\) → \(e\left(\frac{653}{1003}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(7, a) \) \(1\)\(1\)\(e\left(\frac{77}{1003}\right)\)\(e\left(\frac{999}{1003}\right)\)\(e\left(\frac{536}{1003}\right)\)\(e\left(\frac{154}{1003}\right)\)\(e\left(\frac{74}{1003}\right)\)\(e\left(\frac{244}{1003}\right)\)\(e\left(\frac{73}{1003}\right)\)\(e\left(\frac{484}{1003}\right)\)\(e\left(\frac{596}{1003}\right)\)\(e\left(\frac{613}{1003}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(7,a) \;\) at \(\;a = \) e.g. 2