Properties

Label 5586.163
Modulus $5586$
Conductor $931$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,28]))
 
pari: [g,chi] = znchar(Mod(163,5586))
 

Basic properties

Modulus: \(5586\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5586.cx

\(\chi_{5586}(163,\cdot)\) \(\chi_{5586}(235,\cdot)\) \(\chi_{5586}(1033,\cdot)\) \(\chi_{5586}(1759,\cdot)\) \(\chi_{5586}(2557,\cdot)\) \(\chi_{5586}(2629,\cdot)\) \(\chi_{5586}(3355,\cdot)\) \(\chi_{5586}(3427,\cdot)\) \(\chi_{5586}(4153,\cdot)\) \(\chi_{5586}(4225,\cdot)\) \(\chi_{5586}(4951,\cdot)\) \(\chi_{5586}(5023,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 21.21.103818783062189717738091671292152377422379176428329.2

Values on generators

\((3725,4903,4999)\) → \((1,e\left(\frac{10}{21}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5586 }(163, a) \) \(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{17}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5586 }(163,a) \;\) at \(\;a = \) e.g. 2