Properties

Label 8026.53
Modulus $8026$
Conductor $4013$
Order $17$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([16]))
 
pari: [g,chi] = znchar(Mod(53,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.d

\(\chi_{8026}(53,\cdot)\) \(\chi_{8026}(309,\cdot)\) \(\chi_{8026}(325,\cdot)\) \(\chi_{8026}(763,\cdot)\) \(\chi_{8026}(923,\cdot)\) \(\chi_{8026}(1173,\cdot)\) \(\chi_{8026}(1287,\cdot)\) \(\chi_{8026}(2809,\cdot)\) \(\chi_{8026}(3013,\cdot)\) \(\chi_{8026}(3483,\cdot)\) \(\chi_{8026}(4003,\cdot)\) \(\chi_{8026}(4113,\cdot)\) \(\chi_{8026}(4297,\cdot)\) \(\chi_{8026}(4409,\cdot)\) \(\chi_{8026}(5987,\cdot)\) \(\chi_{8026}(7195,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 17 polynomial

Values on generators

\(4015\) → \(e\left(\frac{8}{17}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(53, a) \) \(1\)\(1\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{6}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(53,a) \;\) at \(\;a = \) e.g. 2