Properties

Label 5586.29
Modulus $5586$
Conductor $2793$
Order $126$
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(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,54,119]))
 
pari: [g,chi] = znchar(Mod(29,5586))
 

Basic properties

Modulus: \(5586\)
Conductor: \(2793\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2793}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5586.el

\(\chi_{5586}(29,\cdot)\) \(\chi_{5586}(71,\cdot)\) \(\chi_{5586}(155,\cdot)\) \(\chi_{5586}(281,\cdot)\) \(\chi_{5586}(659,\cdot)\) \(\chi_{5586}(743,\cdot)\) \(\chi_{5586}(827,\cdot)\) \(\chi_{5586}(869,\cdot)\) \(\chi_{5586}(953,\cdot)\) \(\chi_{5586}(1457,\cdot)\) \(\chi_{5586}(1541,\cdot)\) \(\chi_{5586}(1625,\cdot)\) \(\chi_{5586}(1751,\cdot)\) \(\chi_{5586}(1877,\cdot)\) \(\chi_{5586}(2339,\cdot)\) \(\chi_{5586}(2423,\cdot)\) \(\chi_{5586}(2465,\cdot)\) \(\chi_{5586}(2675,\cdot)\) \(\chi_{5586}(3053,\cdot)\) \(\chi_{5586}(3221,\cdot)\) \(\chi_{5586}(3263,\cdot)\) \(\chi_{5586}(3347,\cdot)\) \(\chi_{5586}(3473,\cdot)\) \(\chi_{5586}(3851,\cdot)\) \(\chi_{5586}(3935,\cdot)\) \(\chi_{5586}(4061,\cdot)\) \(\chi_{5586}(4145,\cdot)\) \(\chi_{5586}(4271,\cdot)\) \(\chi_{5586}(4649,\cdot)\) \(\chi_{5586}(4733,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((3725,4903,4999)\) → \((-1,e\left(\frac{3}{7}\right),e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5586 }(29, a) \) \(1\)\(1\)\(e\left(\frac{5}{126}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{109}{126}\right)\)\(e\left(\frac{83}{126}\right)\)\(e\left(\frac{85}{126}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{13}{63}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5586 }(29,a) \;\) at \(\;a = \) e.g. 2