Properties

Label 6017.2016
Modulus $6017$
Conductor $6017$
Order $65$
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(130))
 
M = H._module
 
chi = DirichletCharacter(H, M([104,100]))
 
pari: [g,chi] = znchar(Mod(2016,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(65\)
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.be

\(\chi_{6017}(350,\cdot)\) \(\chi_{6017}(509,\cdot)\) \(\chi_{6017}(784,\cdot)\) \(\chi_{6017}(808,\cdot)\) \(\chi_{6017}(840,\cdot)\) \(\chi_{6017}(900,\cdot)\) \(\chi_{6017}(922,\cdot)\) \(\chi_{6017}(1444,\cdot)\) \(\chi_{6017}(1534,\cdot)\) \(\chi_{6017}(1611,\cdot)\) \(\chi_{6017}(1687,\cdot)\) \(\chi_{6017}(1934,\cdot)\) \(\chi_{6017}(1994,\cdot)\) \(\chi_{6017}(2016,\cdot)\) \(\chi_{6017}(2116,\cdot)\) \(\chi_{6017}(2150,\cdot)\) \(\chi_{6017}(2160,\cdot)\) \(\chi_{6017}(2425,\cdot)\) \(\chi_{6017}(2781,\cdot)\) \(\chi_{6017}(2996,\cdot)\) \(\chi_{6017}(3028,\cdot)\) \(\chi_{6017}(3085,\cdot)\) \(\chi_{6017}(3210,\cdot)\) \(\chi_{6017}(3254,\cdot)\) \(\chi_{6017}(3635,\cdot)\) \(\chi_{6017}(3657,\cdot)\) \(\chi_{6017}(3722,\cdot)\) \(\chi_{6017}(3799,\cdot)\) \(\chi_{6017}(3875,\cdot)\) \(\chi_{6017}(4090,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 65 polynomial

Values on generators

\((3830,2190)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{10}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(2016, a) \) \(1\)\(1\)\(e\left(\frac{37}{65}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{65}\right)\)\(e\left(\frac{58}{65}\right)\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{54}{65}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{7}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(2016,a) \;\) at \(\;a = \) e.g. 2