Properties

Label 6026.19
Modulus $6026$
Conductor $3013$
Order $286$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, base_ring=CyclotomicField(286))
 
M = H._module
 
chi = DirichletCharacter(H, M([195,77]))
 
pari: [g,chi] = znchar(Mod(19,6026))
 

Basic properties

Modulus: \(6026\)
Conductor: \(3013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(286\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3013}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6026.ba

\(\chi_{6026}(19,\cdot)\) \(\chi_{6026}(51,\cdot)\) \(\chi_{6026}(79,\cdot)\) \(\chi_{6026}(149,\cdot)\) \(\chi_{6026}(155,\cdot)\) \(\chi_{6026}(199,\cdot)\) \(\chi_{6026}(217,\cdot)\) \(\chi_{6026}(281,\cdot)\) \(\chi_{6026}(309,\cdot)\) \(\chi_{6026}(313,\cdot)\) \(\chi_{6026}(333,\cdot)\) \(\chi_{6026}(341,\cdot)\) \(\chi_{6026}(411,\cdot)\) \(\chi_{6026}(425,\cdot)\) \(\chi_{6026}(479,\cdot)\) \(\chi_{6026}(543,\cdot)\) \(\chi_{6026}(571,\cdot)\) \(\chi_{6026}(595,\cdot)\) \(\chi_{6026}(603,\cdot)\) \(\chi_{6026}(687,\cdot)\) \(\chi_{6026}(723,\cdot)\) \(\chi_{6026}(741,\cdot)\) \(\chi_{6026}(747,\cdot)\) \(\chi_{6026}(833,\cdot)\) \(\chi_{6026}(865,\cdot)\) \(\chi_{6026}(935,\cdot)\) \(\chi_{6026}(941,\cdot)\) \(\chi_{6026}(985,\cdot)\) \(\chi_{6026}(1003,\cdot)\) \(\chi_{6026}(1009,\cdot)\) ...

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: $\Q(\zeta_{143})$
Fixed field: Number field defined by a degree 286 polynomial (not computed)

Values on generators

\((787,4325)\) → \((e\left(\frac{15}{22}\right),e\left(\frac{7}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6026 }(19, a) \) \(1\)\(1\)\(e\left(\frac{42}{143}\right)\)\(e\left(\frac{19}{286}\right)\)\(e\left(\frac{229}{286}\right)\)\(e\left(\frac{84}{143}\right)\)\(e\left(\frac{61}{286}\right)\)\(e\left(\frac{56}{143}\right)\)\(e\left(\frac{103}{286}\right)\)\(e\left(\frac{50}{143}\right)\)\(e\left(\frac{93}{143}\right)\)\(e\left(\frac{27}{286}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6026 }(19,a) \;\) at \(\;a = \) e.g. 2