Properties

Label 6026.9
Modulus $6026$
Conductor $3013$
Order $715$
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(1430))
 
M = H._module
 
chi = DirichletCharacter(H, M([650,154]))
 
pari: [g,chi] = znchar(Mod(9,6026))
 

Basic properties

Modulus: \(6026\)
Conductor: \(3013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(715\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3013}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6026.bc

\(\chi_{6026}(3,\cdot)\) \(\chi_{6026}(9,\cdot)\) \(\chi_{6026}(13,\cdot)\) \(\chi_{6026}(25,\cdot)\) \(\chi_{6026}(27,\cdot)\) \(\chi_{6026}(35,\cdot)\) \(\chi_{6026}(41,\cdot)\) \(\chi_{6026}(49,\cdot)\) \(\chi_{6026}(55,\cdot)\) \(\chi_{6026}(59,\cdot)\) \(\chi_{6026}(75,\cdot)\) \(\chi_{6026}(77,\cdot)\) \(\chi_{6026}(81,\cdot)\) \(\chi_{6026}(101,\cdot)\) \(\chi_{6026}(105,\cdot)\) \(\chi_{6026}(117,\cdot)\) \(\chi_{6026}(121,\cdot)\) \(\chi_{6026}(123,\cdot)\) \(\chi_{6026}(147,\cdot)\) \(\chi_{6026}(151,\cdot)\) \(\chi_{6026}(165,\cdot)\) \(\chi_{6026}(167,\cdot)\) \(\chi_{6026}(169,\cdot)\) \(\chi_{6026}(177,\cdot)\) \(\chi_{6026}(179,\cdot)\) \(\chi_{6026}(225,\cdot)\) \(\chi_{6026}(233,\cdot)\) \(\chi_{6026}(239,\cdot)\) \(\chi_{6026}(265,\cdot)\) \(\chi_{6026}(269,\cdot)\) ...

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: $\Q(\zeta_{715})$
Fixed field: Number field defined by a degree 715 polynomial (not computed)

Values on generators

\((787,4325)\) → \((e\left(\frac{5}{11}\right),e\left(\frac{7}{65}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6026 }(9, a) \) \(1\)\(1\)\(e\left(\frac{19}{715}\right)\)\(e\left(\frac{292}{715}\right)\)\(e\left(\frac{697}{715}\right)\)\(e\left(\frac{38}{715}\right)\)\(e\left(\frac{87}{715}\right)\)\(e\left(\frac{216}{715}\right)\)\(e\left(\frac{311}{715}\right)\)\(e\left(\frac{581}{715}\right)\)\(e\left(\frac{84}{143}\right)\)\(e\left(\frac{1}{715}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6026 }(9,a) \;\) at \(\;a = \) e.g. 2