Properties

Modulus 55
Conductor 5
Order 4
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 55.f

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(55)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,0]))
 
pari: [g,chi] = znchar(Mod(23,55))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 55
Conductor = 5
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 55.f
Orbit index = 6

Galois orbit

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

\(\chi_{55}(12,\cdot)\) \(\chi_{55}(23,\cdot)\)

Values on generators

\((12,46)\) → \((-i,1)\)

Values

-112346789121314
\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(-i\)\(i\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 55 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{55}(23,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(23,r) e\left(\frac{2r}{55}\right) = 1.9021130326+-1.1755705046i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 55 }(23,·),\chi_{ 55 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{55}(23,\cdot),\chi_{55}(1,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(23,r) \chi_{55}(1,1-r) = -9 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 55 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{55}(23,·)) = \sum_{r \in \Z/55\Z} \chi_{55}(23,r) e\left(\frac{1 r + 2 r^{-1}}{55}\right) = 0.2072670016+0.2072670016i \)