Properties

Label 8003.10
Modulus $8003$
Conductor $8003$
Order $975$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8003, base_ring=CyclotomicField(1950))
 
M = H._module
 
chi = DirichletCharacter(H, M([1800,416]))
 
pari: [g,chi] = znchar(Mod(10,8003))
 

Basic properties

Modulus: \(8003\)
Conductor: \(8003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(975\)
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 8003.cm

\(\chi_{8003}(10,\cdot)\) \(\chi_{8003}(36,\cdot)\) \(\chi_{8003}(42,\cdot)\) \(\chi_{8003}(47,\cdot)\) \(\chi_{8003}(49,\cdot)\) \(\chi_{8003}(69,\cdot)\) \(\chi_{8003}(95,\cdot)\) \(\chi_{8003}(97,\cdot)\) \(\chi_{8003}(99,\cdot)\) \(\chi_{8003}(100,\cdot)\) \(\chi_{8003}(116,\cdot)\) \(\chi_{8003}(121,\cdot)\) \(\chi_{8003}(169,\cdot)\) \(\chi_{8003}(172,\cdot)\) \(\chi_{8003}(187,\cdot)\) \(\chi_{8003}(206,\cdot)\) \(\chi_{8003}(225,\cdot)\) \(\chi_{8003}(248,\cdot)\) \(\chi_{8003}(254,\cdot)\) \(\chi_{8003}(289,\cdot)\) \(\chi_{8003}(307,\cdot)\) \(\chi_{8003}(312,\cdot)\) \(\chi_{8003}(333,\cdot)\) \(\chi_{8003}(342,\cdot)\) \(\chi_{8003}(360,\cdot)\) \(\chi_{8003}(364,\cdot)\) \(\chi_{8003}(399,\cdot)\) \(\chi_{8003}(418,\cdot)\) \(\chi_{8003}(439,\cdot)\) \(\chi_{8003}(440,\cdot)\) ...

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: $\Q(\zeta_{975})$
Fixed field: Number field defined by a degree 975 polynomial (not computed)

Values on generators

\((4984,7103)\) → \((e\left(\frac{12}{13}\right),e\left(\frac{16}{75}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8003 }(10, a) \) \(1\)\(1\)\(e\left(\frac{167}{195}\right)\)\(e\left(\frac{316}{325}\right)\)\(e\left(\frac{139}{195}\right)\)\(e\left(\frac{271}{975}\right)\)\(e\left(\frac{808}{975}\right)\)\(e\left(\frac{211}{975}\right)\)\(e\left(\frac{37}{65}\right)\)\(e\left(\frac{307}{325}\right)\)\(e\left(\frac{131}{975}\right)\)\(e\left(\frac{577}{975}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8003 }(10,a) \;\) at \(\;a = \) e.g. 2