Properties

Label 6012.959
Modulus $6012$
Conductor $6012$
Order $498$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6012, base_ring=CyclotomicField(498))
 
M = H._module
 
chi = DirichletCharacter(H, M([249,415,12]))
 
pari: [g,chi] = znchar(Mod(959,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(6012\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(498\)
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 6012.ba

\(\chi_{6012}(11,\cdot)\) \(\chi_{6012}(47,\cdot)\) \(\chi_{6012}(191,\cdot)\) \(\chi_{6012}(203,\cdot)\) \(\chi_{6012}(239,\cdot)\) \(\chi_{6012}(263,\cdot)\) \(\chi_{6012}(275,\cdot)\) \(\chi_{6012}(299,\cdot)\) \(\chi_{6012}(311,\cdot)\) \(\chi_{6012}(383,\cdot)\) \(\chi_{6012}(419,\cdot)\) \(\chi_{6012}(455,\cdot)\) \(\chi_{6012}(491,\cdot)\) \(\chi_{6012}(515,\cdot)\) \(\chi_{6012}(551,\cdot)\) \(\chi_{6012}(563,\cdot)\) \(\chi_{6012}(599,\cdot)\) \(\chi_{6012}(623,\cdot)\) \(\chi_{6012}(671,\cdot)\) \(\chi_{6012}(695,\cdot)\) \(\chi_{6012}(731,\cdot)\) \(\chi_{6012}(743,\cdot)\) \(\chi_{6012}(767,\cdot)\) \(\chi_{6012}(815,\cdot)\) \(\chi_{6012}(839,\cdot)\) \(\chi_{6012}(851,\cdot)\) \(\chi_{6012}(911,\cdot)\) \(\chi_{6012}(923,\cdot)\) \(\chi_{6012}(947,\cdot)\) \(\chi_{6012}(959,\cdot)\) ...

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: $\Q(\zeta_{249})$
Fixed field: Number field defined by a degree 498 polynomial (not computed)

Values on generators

\((3007,3341,4681)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{2}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 6012 }(959, a) \) \(1\)\(1\)\(e\left(\frac{95}{498}\right)\)\(e\left(\frac{337}{498}\right)\)\(e\left(\frac{2}{249}\right)\)\(e\left(\frac{37}{249}\right)\)\(e\left(\frac{129}{166}\right)\)\(e\left(\frac{149}{166}\right)\)\(e\left(\frac{13}{249}\right)\)\(e\left(\frac{95}{249}\right)\)\(e\left(\frac{223}{498}\right)\)\(e\left(\frac{167}{498}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6012 }(959,a) \;\) at \(\;a = \) e.g. 2