Properties

Label 5586.17
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([63,75,70]))
 
pari: [g,chi] = znchar(Mod(17,5586))
 

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 \(\chi_{2793}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5586.eh

\(\chi_{5586}(17,\cdot)\) \(\chi_{5586}(47,\cdot)\) \(\chi_{5586}(593,\cdot)\) \(\chi_{5586}(605,\cdot)\) \(\chi_{5586}(689,\cdot)\) \(\chi_{5586}(845,\cdot)\) \(\chi_{5586}(1013,\cdot)\) \(\chi_{5586}(1487,\cdot)\) \(\chi_{5586}(1613,\cdot)\) \(\chi_{5586}(1643,\cdot)\) \(\chi_{5586}(1811,\cdot)\) \(\chi_{5586}(2189,\cdot)\) \(\chi_{5586}(2201,\cdot)\) \(\chi_{5586}(2411,\cdot)\) \(\chi_{5586}(2441,\cdot)\) \(\chi_{5586}(2609,\cdot)\) \(\chi_{5586}(2987,\cdot)\) \(\chi_{5586}(2999,\cdot)\) \(\chi_{5586}(3083,\cdot)\) \(\chi_{5586}(3209,\cdot)\) \(\chi_{5586}(3239,\cdot)\) \(\chi_{5586}(3407,\cdot)\) \(\chi_{5586}(3785,\cdot)\) \(\chi_{5586}(3797,\cdot)\) \(\chi_{5586}(3881,\cdot)\) \(\chi_{5586}(4007,\cdot)\) \(\chi_{5586}(4205,\cdot)\) \(\chi_{5586}(4583,\cdot)\) \(\chi_{5586}(4595,\cdot)\) \(\chi_{5586}(4679,\cdot)\) ...

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

\((3725,4903,4999)\) → \((-1,e\left(\frac{25}{42}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5586 }(17, a) \) \(1\)\(1\)\(e\left(\frac{41}{63}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{53}{126}\right)\)\(e\left(\frac{59}{63}\right)\)\(e\left(\frac{29}{126}\right)\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{83}{126}\right)\)\(-1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{41}{63}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5586 }(17,a) \;\) at \(\;a = \) e.g. 2