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Properties

Label 5.2
Modulus $5$
Conductor $5$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content pari:[g,chi] = znchar(Mod(2,5))
 

Basic properties

Modulus: \(5\)
Conductor: \(5\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 5.c

\(\chi_{5}(2,\cdot)\) \(\chi_{5}(3,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: \(\Q(\zeta_{5})\)

Values on generators

\(2\) → \(i\)

Values

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

Gauss sum

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

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 5 }(2,·),\chi_{ 5 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

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

Additional information

This is the first example of a Dirichlet character whose values do not all lie in the field of rational numbers.

This also makes it the first example of a Dirichlet character whose Galois orbit is nontrivial.