Properties

Label 6017.53
Modulus $6017$
Conductor $6017$
Order $1365$
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(2730))
 
M = H._module
 
chi = DirichletCharacter(H, M([1638,1220]))
 
pari: [g,chi] = znchar(Mod(53,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1365\)
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.ci

\(\chi_{6017}(4,\cdot)\) \(\chi_{6017}(15,\cdot)\) \(\chi_{6017}(16,\cdot)\) \(\chi_{6017}(25,\cdot)\) \(\chi_{6017}(36,\cdot)\) \(\chi_{6017}(49,\cdot)\) \(\chi_{6017}(53,\cdot)\) \(\chi_{6017}(60,\cdot)\) \(\chi_{6017}(69,\cdot)\) \(\chi_{6017}(82,\cdot)\) \(\chi_{6017}(86,\cdot)\) \(\chi_{6017}(97,\cdot)\) \(\chi_{6017}(113,\cdot)\) \(\chi_{6017}(115,\cdot)\) \(\chi_{6017}(119,\cdot)\) \(\chi_{6017}(130,\cdot)\) \(\chi_{6017}(135,\cdot)\) \(\chi_{6017}(137,\cdot)\) \(\chi_{6017}(157,\cdot)\) \(\chi_{6017}(158,\cdot)\) \(\chi_{6017}(174,\cdot)\) \(\chi_{6017}(190,\cdot)\) \(\chi_{6017}(191,\cdot)\) \(\chi_{6017}(202,\cdot)\) \(\chi_{6017}(213,\cdot)\) \(\chi_{6017}(214,\cdot)\) \(\chi_{6017}(225,\cdot)\) \(\chi_{6017}(256,\cdot)\) \(\chi_{6017}(269,\cdot)\) \(\chi_{6017}(273,\cdot)\) ...

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: $\Q(\zeta_{1365})$
Fixed field: Number field defined by a degree 1365 polynomial (not computed)

Values on generators

\((3830,2190)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{122}{273}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(53, a) \) \(1\)\(1\)\(e\left(\frac{64}{1365}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{128}{1365}\right)\)\(e\left(\frac{536}{1365}\right)\)\(e\left(\frac{766}{1365}\right)\)\(e\left(\frac{313}{1365}\right)\)\(e\left(\frac{64}{455}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{40}{91}\right)\)\(e\left(\frac{166}{273}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(53,a) \;\) at \(\;a = \) e.g. 2