Properties

Label 8.1
Modulus $8$
Conductor $1$
Order $1$
Real yes
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0]))
 
pari: [g,chi] = znchar(Mod(1,8))
 

Basic properties

Modulus: \(8\)
Conductor: \(1\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{1}(1,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8.a

\(\chi_{8}(1,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((7,5)\) → \((1,1)\)

Values

\(-1\)\(1\)\(3\)\(5\)
\(1\)\(1\)\(1\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 8 }(1,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{8}(1,\cdot)) = \sum_{r\in \Z/8\Z} \chi_{8}(1,r) e\left(\frac{r}{4}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 8 }(1,·),\chi_{ 8 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{8}(1,\cdot),\chi_{8}(1,\cdot)) = \sum_{r\in \Z/8\Z} \chi_{8}(1,r) \chi_{8}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 8 }(1,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{8}(1,·)) = \sum_{r \in \Z/8\Z} \chi_{8}(1,r) e\left(\frac{1 r + 2 r^{-1}}{8}\right) = 0.0 \)