Properties

Label 6008.775
Modulus $6008$
Conductor $3004$
Order $750$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, base_ring=CyclotomicField(750))
 
M = H._module
 
chi = DirichletCharacter(H, M([375,0,499]))
 
pari: [g,chi] = znchar(Mod(775,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(3004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(750\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3004}(775,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6008.ci

\(\chi_{6008}(15,\cdot)\) \(\chi_{6008}(31,\cdot)\) \(\chi_{6008}(39,\cdot)\) \(\chi_{6008}(55,\cdot)\) \(\chi_{6008}(63,\cdot)\) \(\chi_{6008}(79,\cdot)\) \(\chi_{6008}(103,\cdot)\) \(\chi_{6008}(135,\cdot)\) \(\chi_{6008}(143,\cdot)\) \(\chi_{6008}(159,\cdot)\) \(\chi_{6008}(175,\cdot)\) \(\chi_{6008}(231,\cdot)\) \(\chi_{6008}(263,\cdot)\) \(\chi_{6008}(279,\cdot)\) \(\chi_{6008}(335,\cdot)\) \(\chi_{6008}(351,\cdot)\) \(\chi_{6008}(375,\cdot)\) \(\chi_{6008}(431,\cdot)\) \(\chi_{6008}(487,\cdot)\) \(\chi_{6008}(495,\cdot)\) \(\chi_{6008}(591,\cdot)\) \(\chi_{6008}(599,\cdot)\) \(\chi_{6008}(607,\cdot)\) \(\chi_{6008}(623,\cdot)\) \(\chi_{6008}(647,\cdot)\) \(\chi_{6008}(711,\cdot)\) \(\chi_{6008}(735,\cdot)\) \(\chi_{6008}(775,\cdot)\) \(\chi_{6008}(839,\cdot)\) \(\chi_{6008}(847,\cdot)\) ...

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: $\Q(\zeta_{375})$
Fixed field: Number field defined by a degree 750 polynomial (not computed)

Values on generators

\((1503,3005,1505)\) → \((-1,1,e\left(\frac{499}{750}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(775, a) \) \(1\)\(1\)\(e\left(\frac{62}{375}\right)\)\(e\left(\frac{257}{375}\right)\)\(e\left(\frac{79}{125}\right)\)\(e\left(\frac{124}{375}\right)\)\(e\left(\frac{16}{75}\right)\)\(e\left(\frac{371}{375}\right)\)\(e\left(\frac{319}{375}\right)\)\(e\left(\frac{671}{750}\right)\)\(e\left(\frac{607}{750}\right)\)\(e\left(\frac{299}{375}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(775,a) \;\) at \(\;a = \) e.g. 2