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Properties

Label 8.7
Modulus $8$
Conductor $4$
Order $2$
Real yes
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(7,8))
 

Basic properties

Modulus: \(8\)
Conductor: \(4\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{4}(3,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 8.c

\(\chi_{8}(7,\cdot)\)

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

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-1}) \)

Values on generators

\((7,5)\) → \((-1,1)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)
\( \chi_{ 8 }(7, a) \) \(-1\)\(1\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 8 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 8 }(7,·),\chi_{ 8 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 8 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2

Additional information

This is the first example of a Dirichlet character that is not minimal.