Properties

Label 8014.17
Modulus $8014$
Conductor $4007$
Order $4006$
Real no
Primitive no
Minimal yes
Parity odd

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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([581]))
 
pari: [g,chi] = znchar(Mod(17,8014))
 

Basic properties

Modulus: \(8014\)
Conductor: \(4007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4006\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4007}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8014.d

\(\chi_{8014}(5,\cdot)\) \(\chi_{8014}(11,\cdot)\) \(\chi_{8014}(15,\cdot)\) \(\chi_{8014}(17,\cdot)\) \(\chi_{8014}(19,\cdot)\) \(\chi_{8014}(31,\cdot)\) \(\chi_{8014}(33,\cdot)\) \(\chi_{8014}(35,\cdot)\) \(\chi_{8014}(41,\cdot)\) \(\chi_{8014}(45,\cdot)\) \(\chi_{8014}(47,\cdot)\) \(\chi_{8014}(51,\cdot)\) \(\chi_{8014}(53,\cdot)\) \(\chi_{8014}(57,\cdot)\) \(\chi_{8014}(65,\cdot)\) \(\chi_{8014}(67,\cdot)\) \(\chi_{8014}(77,\cdot)\) \(\chi_{8014}(83,\cdot)\) \(\chi_{8014}(93,\cdot)\) \(\chi_{8014}(97,\cdot)\) \(\chi_{8014}(99,\cdot)\) \(\chi_{8014}(103,\cdot)\) \(\chi_{8014}(105,\cdot)\) \(\chi_{8014}(107,\cdot)\) \(\chi_{8014}(115,\cdot)\) \(\chi_{8014}(119,\cdot)\) \(\chi_{8014}(123,\cdot)\) \(\chi_{8014}(125,\cdot)\) \(\chi_{8014}(127,\cdot)\) \(\chi_{8014}(131,\cdot)\) ...

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 4006 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{581}{4006}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8014 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{1372}{2003}\right)\)\(e\left(\frac{581}{4006}\right)\)\(e\left(\frac{1131}{2003}\right)\)\(e\left(\frac{741}{2003}\right)\)\(e\left(\frac{141}{4006}\right)\)\(e\left(\frac{267}{2003}\right)\)\(e\left(\frac{3325}{4006}\right)\)\(e\left(\frac{1057}{4006}\right)\)\(e\left(\frac{2577}{4006}\right)\)\(e\left(\frac{500}{2003}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8014 }(17,a) \;\) at \(\;a = \) e.g. 2