Properties

Label 6019.7
Modulus $6019$
Conductor $6019$
Order $924$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6019, base_ring=CyclotomicField(924))
 
M = H._module
 
chi = DirichletCharacter(H, M([847,306]))
 
pari: [g,chi] = znchar(Mod(7,6019))
 

Basic properties

Modulus: \(6019\)
Conductor: \(6019\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(924\)
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 6019.en

\(\chi_{6019}(7,\cdot)\) \(\chi_{6019}(71,\cdot)\) \(\chi_{6019}(80,\cdot)\) \(\chi_{6019}(106,\cdot)\) \(\chi_{6019}(150,\cdot)\) \(\chi_{6019}(180,\cdot)\) \(\chi_{6019}(184,\cdot)\) \(\chi_{6019}(193,\cdot)\) \(\chi_{6019}(197,\cdot)\) \(\chi_{6019}(201,\cdot)\) \(\chi_{6019}(219,\cdot)\) \(\chi_{6019}(254,\cdot)\) \(\chi_{6019}(314,\cdot)\) \(\chi_{6019}(319,\cdot)\) \(\chi_{6019}(340,\cdot)\) \(\chi_{6019}(366,\cdot)\) \(\chi_{6019}(379,\cdot)\) \(\chi_{6019}(397,\cdot)\) \(\chi_{6019}(405,\cdot)\) \(\chi_{6019}(414,\cdot)\) \(\chi_{6019}(470,\cdot)\) \(\chi_{6019}(475,\cdot)\) \(\chi_{6019}(592,\cdot)\) \(\chi_{6019}(613,\cdot)\) \(\chi_{6019}(643,\cdot)\) \(\chi_{6019}(656,\cdot)\) \(\chi_{6019}(682,\cdot)\) \(\chi_{6019}(700,\cdot)\) \(\chi_{6019}(717,\cdot)\) \(\chi_{6019}(765,\cdot)\) ...

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: $\Q(\zeta_{924})$
Fixed field: Number field defined by a degree 924 polynomial (not computed)

Values on generators

\((2316,1392)\) → \((e\left(\frac{11}{12}\right),e\left(\frac{51}{154}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6019 }(7, a) \) \(1\)\(1\)\(e\left(\frac{163}{924}\right)\)\(e\left(\frac{461}{462}\right)\)\(e\left(\frac{163}{462}\right)\)\(e\left(\frac{199}{308}\right)\)\(e\left(\frac{23}{132}\right)\)\(e\left(\frac{695}{924}\right)\)\(e\left(\frac{163}{308}\right)\)\(e\left(\frac{230}{231}\right)\)\(e\left(\frac{190}{231}\right)\)\(e\left(\frac{607}{924}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6019 }(7,a) \;\) at \(\;a = \) e.g. 2