Properties

Label 6008.4557
Modulus $6008$
Conductor $6008$
Order $50$
Real no
Primitive yes
Minimal yes
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([0,25,22]))
 
pari: [g,chi] = znchar(Mod(4557,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(6008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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 6008.bl

\(\chi_{6008}(53,\cdot)\) \(\chi_{6008}(117,\cdot)\) \(\chi_{6008}(325,\cdot)\) \(\chi_{6008}(485,\cdot)\) \(\chi_{6008}(1461,\cdot)\) \(\chi_{6008}(2205,\cdot)\) \(\chi_{6008}(3197,\cdot)\) \(\chi_{6008}(3485,\cdot)\) \(\chi_{6008}(3917,\cdot)\) \(\chi_{6008}(4205,\cdot)\) \(\chi_{6008}(4221,\cdot)\) \(\chi_{6008}(4421,\cdot)\) \(\chi_{6008}(4557,\cdot)\) \(\chi_{6008}(4677,\cdot)\) \(\chi_{6008}(4685,\cdot)\) \(\chi_{6008}(4981,\cdot)\) \(\chi_{6008}(5005,\cdot)\) \(\chi_{6008}(5605,\cdot)\) \(\chi_{6008}(5677,\cdot)\) \(\chi_{6008}(5813,\cdot)\)

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

\((1503,3005,1505)\) → \((1,-1,e\left(\frac{11}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(4557, a) \) \(1\)\(1\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{19}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(4557,a) \;\) at \(\;a = \) e.g. 2