Properties

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

Related objects

Learn more about

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

Basic properties

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

Galois orbit 5.c

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

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

\(2\) → \(i\)

Values

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

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 5 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{5}(2,\cdot)) = \sum_{r\in \Z/5\Z} \chi_{5}(2,r) e\left(\frac{2r}{5}\right) = 1.9021130326+1.1755705046i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 5 }(2,·),\chi_{ 5 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{5}(2,\cdot),\chi_{5}(1,\cdot)) = \sum_{r\in \Z/5\Z} \chi_{5}(2,r) \chi_{5}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 5 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{5}(2,·)) = \sum_{r \in \Z/5\Z} \chi_{5}(2,r) e\left(\frac{1 r + 2 r^{-1}}{5}\right) = 1.1755705046+-1.1755705046i \)

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.