Properties

Label 10009.17
Modulus $10009$
Conductor $10009$
Order $5004$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10009, base_ring=CyclotomicField(5004))
 
M = H._module
 
chi = DirichletCharacter(H, M([3985]))
 
pari: [g,chi] = znchar(Mod(17,10009))
 

Basic properties

Modulus: \(10009\)
Conductor: \(10009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(5004\)
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 10009.w

\(\chi_{10009}(3,\cdot)\) \(\chi_{10009}(10,\cdot)\) \(\chi_{10009}(12,\cdot)\) \(\chi_{10009}(17,\cdot)\) \(\chi_{10009}(18,\cdot)\) \(\chi_{10009}(23,\cdot)\) \(\chi_{10009}(40,\cdot)\) \(\chi_{10009}(48,\cdot)\) \(\chi_{10009}(50,\cdot)\) \(\chi_{10009}(58,\cdot)\) \(\chi_{10009}(61,\cdot)\) \(\chi_{10009}(65,\cdot)\) \(\chi_{10009}(67,\cdot)\) \(\chi_{10009}(68,\cdot)\) \(\chi_{10009}(72,\cdot)\) \(\chi_{10009}(75,\cdot)\) \(\chi_{10009}(78,\cdot)\) \(\chi_{10009}(85,\cdot)\) \(\chi_{10009}(87,\cdot)\) \(\chi_{10009}(90,\cdot)\) \(\chi_{10009}(92,\cdot)\) \(\chi_{10009}(103,\cdot)\) \(\chi_{10009}(115,\cdot)\) \(\chi_{10009}(117,\cdot)\) \(\chi_{10009}(121,\cdot)\) \(\chi_{10009}(135,\cdot)\) \(\chi_{10009}(137,\cdot)\) \(\chi_{10009}(139,\cdot)\) \(\chi_{10009}(146,\cdot)\) \(\chi_{10009}(149,\cdot)\) ...

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: $\Q(\zeta_{5004})$
Fixed field: Number field defined by a degree 5004 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{3985}{5004}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 10009 }(17, a) \) \(1\)\(1\)\(e\left(\frac{797}{834}\right)\)\(e\left(\frac{1985}{2502}\right)\)\(e\left(\frac{380}{417}\right)\)\(e\left(\frac{239}{1251}\right)\)\(e\left(\frac{937}{1251}\right)\)\(e\left(\frac{63}{556}\right)\)\(e\left(\frac{241}{278}\right)\)\(e\left(\frac{734}{1251}\right)\)\(e\left(\frac{367}{2502}\right)\)\(e\left(\frac{3985}{5004}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 10009 }(17,a) \;\) at \(\;a = \) e.g. 2