Properties

Label 8026.831
Modulus $8026$
Conductor $4013$
Order $34$
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([5]))
 
pari: [g,chi] = znchar(Mod(831,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(831,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.e

\(\chi_{8026}(831,\cdot)\) \(\chi_{8026}(2039,\cdot)\) \(\chi_{8026}(3617,\cdot)\) \(\chi_{8026}(3729,\cdot)\) \(\chi_{8026}(3913,\cdot)\) \(\chi_{8026}(4023,\cdot)\) \(\chi_{8026}(4543,\cdot)\) \(\chi_{8026}(5013,\cdot)\) \(\chi_{8026}(5217,\cdot)\) \(\chi_{8026}(6739,\cdot)\) \(\chi_{8026}(6853,\cdot)\) \(\chi_{8026}(7103,\cdot)\) \(\chi_{8026}(7263,\cdot)\) \(\chi_{8026}(7701,\cdot)\) \(\chi_{8026}(7717,\cdot)\) \(\chi_{8026}(7973,\cdot)\)

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 34 polynomial

Values on generators

\(4015\) → \(e\left(\frac{5}{34}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(831, a) \) \(1\)\(1\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{25}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(831,a) \;\) at \(\;a = \) e.g. 2