Properties

Label 5082.439
Modulus $5082$
Conductor $847$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5082, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,51]))
 
pari: [g,chi] = znchar(Mod(439,5082))
 

Basic properties

Modulus: \(5082\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(439,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5082.br

\(\chi_{5082}(439,\cdot)\) \(\chi_{5082}(703,\cdot)\) \(\chi_{5082}(901,\cdot)\) \(\chi_{5082}(1165,\cdot)\) \(\chi_{5082}(1363,\cdot)\) \(\chi_{5082}(1627,\cdot)\) \(\chi_{5082}(1825,\cdot)\) \(\chi_{5082}(2089,\cdot)\) \(\chi_{5082}(2287,\cdot)\) \(\chi_{5082}(2551,\cdot)\) \(\chi_{5082}(2749,\cdot)\) \(\chi_{5082}(3013,\cdot)\) \(\chi_{5082}(3211,\cdot)\) \(\chi_{5082}(3475,\cdot)\) \(\chi_{5082}(3673,\cdot)\) \(\chi_{5082}(3937,\cdot)\) \(\chi_{5082}(4135,\cdot)\) \(\chi_{5082}(4399,\cdot)\) \(\chi_{5082}(4861,\cdot)\) \(\chi_{5082}(5059,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((3389,4357,2059)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5082 }(439, a) \) \(1\)\(1\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5082 }(439,a) \;\) at \(\;a = \) e.g. 2