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Properties

Label 8.5
Modulus 88
Conductor 88
Order 22
Real yes
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(5,8))
 

Kronecker symbol representation

sage: kronecker_character(8)
 
pari: znchartokronecker(g,chi)
 

(8∙)\displaystyle\left(\frac{8}{\bullet}\right)

Basic properties

Modulus: 88
Conductor: 88
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 22
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8.b

χ8(5,⋅)\chi_{8}(5,\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\Q
Fixed field: Q(2)\Q(\sqrt{2})

Values on generators

(7,5)(7,5) → (1,−1)(1,-1)

Values

aa −1-1113355
χ8(5,a) \chi_{ 8 }(5, a) 1111−1-1−1-1
sage: chi.jacobi_sum(n)
 
χ8(5,a)   \chi_{ 8 }(5,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ8(5,⋅))   \tau_{ a }( \chi_{ 8 }(5,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ8(5,⋅),χ8(n,⋅))   J(\chi_{ 8 }(5,·),\chi_{ 8 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ8(5,⋅))  K(a,b,\chi_{ 8 }(5,·)) \; at   a,b=\; a,b = e.g. 1,2