Properties

Label 6008.4007
Modulus $6008$
Conductor $3004$
Order $50$
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(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([25,0,49]))
 
pari: [g,chi] = znchar(Mod(4007,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(3004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3004}(1003,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6008.bi

\(\chi_{6008}(799,\cdot)\) \(\chi_{6008}(1543,\cdot)\) \(\chi_{6008}(2519,\cdot)\) \(\chi_{6008}(2679,\cdot)\) \(\chi_{6008}(2887,\cdot)\) \(\chi_{6008}(2951,\cdot)\) \(\chi_{6008}(3199,\cdot)\) \(\chi_{6008}(3335,\cdot)\) \(\chi_{6008}(3407,\cdot)\) \(\chi_{6008}(4007,\cdot)\) \(\chi_{6008}(4031,\cdot)\) \(\chi_{6008}(4327,\cdot)\) \(\chi_{6008}(4335,\cdot)\) \(\chi_{6008}(4455,\cdot)\) \(\chi_{6008}(4591,\cdot)\) \(\chi_{6008}(4791,\cdot)\) \(\chi_{6008}(4807,\cdot)\) \(\chi_{6008}(5095,\cdot)\) \(\chi_{6008}(5527,\cdot)\) \(\chi_{6008}(5815,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((1503,3005,1505)\) → \((-1,1,e\left(\frac{49}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(4007, a) \) \(1\)\(1\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{24}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(4007,a) \;\) at \(\;a = \) e.g. 2