Properties

Label 8003.423
Modulus $8003$
Conductor $8003$
Order $150$
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(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,38]))
 
pari: [g,chi] = znchar(Mod(423,8003))
 

Basic properties

Modulus: \(8003\)
Conductor: \(8003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(150\)
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.bt

\(\chi_{8003}(423,\cdot)\) \(\chi_{8003}(635,\cdot)\) \(\chi_{8003}(741,\cdot)\) \(\chi_{8003}(794,\cdot)\) \(\chi_{8003}(900,\cdot)\) \(\chi_{8003}(953,\cdot)\) \(\chi_{8003}(1006,\cdot)\) \(\chi_{8003}(1112,\cdot)\) \(\chi_{8003}(1218,\cdot)\) \(\chi_{8003}(1324,\cdot)\) \(\chi_{8003}(1377,\cdot)\) \(\chi_{8003}(1695,\cdot)\) \(\chi_{8003}(1854,\cdot)\) \(\chi_{8003}(1907,\cdot)\) \(\chi_{8003}(2066,\cdot)\) \(\chi_{8003}(2119,\cdot)\) \(\chi_{8003}(2172,\cdot)\) \(\chi_{8003}(2437,\cdot)\) \(\chi_{8003}(2490,\cdot)\) \(\chi_{8003}(2755,\cdot)\) \(\chi_{8003}(2808,\cdot)\) \(\chi_{8003}(2914,\cdot)\) \(\chi_{8003}(3391,\cdot)\) \(\chi_{8003}(3762,\cdot)\) \(\chi_{8003}(3815,\cdot)\) \(\chi_{8003}(4239,\cdot)\) \(\chi_{8003}(4610,\cdot)\) \(\chi_{8003}(4769,\cdot)\) \(\chi_{8003}(4875,\cdot)\) \(\chi_{8003}(5458,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\((4984,7103)\) → \((-1,e\left(\frac{19}{75}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8003 }(423, a) \) \(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{131}{150}\right)\)\(e\left(\frac{19}{75}\right)\)\(e\left(\frac{73}{75}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{8}{75}\right)\)\(e\left(\frac{61}{75}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8003 }(423,a) \;\) at \(\;a = \) e.g. 2