Properties

Label 6013.55
Modulus $6013$
Conductor $6013$
Order $858$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, base_ring=CyclotomicField(858))
 
M = H._module
 
chi = DirichletCharacter(H, M([429,749]))
 
pari: [g,chi] = znchar(Mod(55,6013))
 

Basic properties

Modulus: \(6013\)
Conductor: \(6013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(858\)
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 6013.cw

\(\chi_{6013}(55,\cdot)\) \(\chi_{6013}(62,\cdot)\) \(\chi_{6013}(76,\cdot)\) \(\chi_{6013}(83,\cdot)\) \(\chi_{6013}(90,\cdot)\) \(\chi_{6013}(97,\cdot)\) \(\chi_{6013}(118,\cdot)\) \(\chi_{6013}(139,\cdot)\) \(\chi_{6013}(160,\cdot)\) \(\chi_{6013}(174,\cdot)\) \(\chi_{6013}(237,\cdot)\) \(\chi_{6013}(251,\cdot)\) \(\chi_{6013}(293,\cdot)\) \(\chi_{6013}(328,\cdot)\) \(\chi_{6013}(349,\cdot)\) \(\chi_{6013}(363,\cdot)\) \(\chi_{6013}(377,\cdot)\) \(\chi_{6013}(419,\cdot)\) \(\chi_{6013}(447,\cdot)\) \(\chi_{6013}(454,\cdot)\) \(\chi_{6013}(489,\cdot)\) \(\chi_{6013}(517,\cdot)\) \(\chi_{6013}(538,\cdot)\) \(\chi_{6013}(545,\cdot)\) \(\chi_{6013}(573,\cdot)\) \(\chi_{6013}(580,\cdot)\) \(\chi_{6013}(594,\cdot)\) \(\chi_{6013}(601,\cdot)\) \(\chi_{6013}(615,\cdot)\) \(\chi_{6013}(650,\cdot)\) ...

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: $\Q(\zeta_{429})$
Fixed field: Number field defined by a degree 858 polynomial (not computed)

Values on generators

\((5155,3438)\) → \((-1,e\left(\frac{749}{858}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6013 }(55, a) \) \(1\)\(1\)\(e\left(\frac{749}{858}\right)\)\(e\left(\frac{164}{429}\right)\)\(e\left(\frac{320}{429}\right)\)\(e\left(\frac{439}{858}\right)\)\(e\left(\frac{73}{286}\right)\)\(e\left(\frac{177}{286}\right)\)\(e\left(\frac{328}{429}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{239}{286}\right)\)\(e\left(\frac{5}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6013 }(55,a) \;\) at \(\;a = \) e.g. 2