Properties

Label 8014.13
Modulus $8014$
Conductor $4007$
Order $2003$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8014, base_ring=CyclotomicField(4006))
 
M = H._module
 
chi = DirichletCharacter(H, M([718]))
 
pari: [g,chi] = znchar(Mod(13,8014))
 

Basic properties

Modulus: \(8014\)
Conductor: \(4007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2003\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4007}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8014.c

\(\chi_{8014}(3,\cdot)\) \(\chi_{8014}(7,\cdot)\) \(\chi_{8014}(9,\cdot)\) \(\chi_{8014}(13,\cdot)\) \(\chi_{8014}(21,\cdot)\) \(\chi_{8014}(23,\cdot)\) \(\chi_{8014}(25,\cdot)\) \(\chi_{8014}(27,\cdot)\) \(\chi_{8014}(29,\cdot)\) \(\chi_{8014}(37,\cdot)\) \(\chi_{8014}(39,\cdot)\) \(\chi_{8014}(43,\cdot)\) \(\chi_{8014}(49,\cdot)\) \(\chi_{8014}(55,\cdot)\) \(\chi_{8014}(59,\cdot)\) \(\chi_{8014}(61,\cdot)\) \(\chi_{8014}(63,\cdot)\) \(\chi_{8014}(69,\cdot)\) \(\chi_{8014}(71,\cdot)\) \(\chi_{8014}(73,\cdot)\) \(\chi_{8014}(75,\cdot)\) \(\chi_{8014}(79,\cdot)\) \(\chi_{8014}(81,\cdot)\) \(\chi_{8014}(85,\cdot)\) \(\chi_{8014}(87,\cdot)\) \(\chi_{8014}(89,\cdot)\) \(\chi_{8014}(91,\cdot)\) \(\chi_{8014}(95,\cdot)\) \(\chi_{8014}(101,\cdot)\) \(\chi_{8014}(109,\cdot)\) ...

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: $\Q(\zeta_{2003})$
Fixed field: Number field defined by a degree 2003 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{359}{2003}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8014 }(13, a) \) \(1\)\(1\)\(e\left(\frac{1237}{2003}\right)\)\(e\left(\frac{359}{2003}\right)\)\(e\left(\frac{1246}{2003}\right)\)\(e\left(\frac{471}{2003}\right)\)\(e\left(\frac{949}{2003}\right)\)\(e\left(\frac{1378}{2003}\right)\)\(e\left(\frac{1596}{2003}\right)\)\(e\left(\frac{267}{2003}\right)\)\(e\left(\frac{596}{2003}\right)\)\(e\left(\frac{480}{2003}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8014 }(13,a) \;\) at \(\;a = \) e.g. 2