Properties

Label 6017.5598
Modulus $6017$
Conductor $6017$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6017, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,1]))
 
pari: [g,chi] = znchar(Mod(5598,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 6017.bk

\(\chi_{6017}(692,\cdot)\) \(\chi_{6017}(769,\cdot)\) \(\chi_{6017}(1594,\cdot)\) \(\chi_{6017}(1957,\cdot)\) \(\chi_{6017}(2067,\cdot)\) \(\chi_{6017}(2177,\cdot)\) \(\chi_{6017}(2496,\cdot)\) \(\chi_{6017}(2518,\cdot)\) \(\chi_{6017}(2606,\cdot)\) \(\chi_{6017}(2639,\cdot)\) \(\chi_{6017}(2837,\cdot)\) \(\chi_{6017}(2980,\cdot)\) \(\chi_{6017}(3101,\cdot)\) \(\chi_{6017}(3365,\cdot)\) \(\chi_{6017}(3596,\cdot)\) \(\chi_{6017}(4080,\cdot)\) \(\chi_{6017}(4322,\cdot)\) \(\chi_{6017}(4355,\cdot)\) \(\chi_{6017}(4949,\cdot)\) \(\chi_{6017}(4982,\cdot)\) \(\chi_{6017}(5334,\cdot)\) \(\chi_{6017}(5576,\cdot)\) \(\chi_{6017}(5598,\cdot)\) \(\chi_{6017}(5818,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((3830,2190)\) → \((-1,e\left(\frac{1}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(5598, a) \) \(1\)\(1\)\(e\left(\frac{20}{39}\right)\)\(-1\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{7}{13}\right)\)\(1\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{41}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(5598,a) \;\) at \(\;a = \) e.g. 2