Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([170,126]))
 
Copy content gp:[g,chi] = znchar(Mod(284, 539))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("539.284");
 

Basic properties

Modulus: \(539\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(539\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 539.bc

\(\chi_{539}(4,\cdot)\) \(\chi_{539}(9,\cdot)\) \(\chi_{539}(16,\cdot)\) \(\chi_{539}(25,\cdot)\) \(\chi_{539}(37,\cdot)\) \(\chi_{539}(53,\cdot)\) \(\chi_{539}(58,\cdot)\) \(\chi_{539}(60,\cdot)\) \(\chi_{539}(81,\cdot)\) \(\chi_{539}(86,\cdot)\) \(\chi_{539}(93,\cdot)\) \(\chi_{539}(102,\cdot)\) \(\chi_{539}(114,\cdot)\) \(\chi_{539}(130,\cdot)\) \(\chi_{539}(135,\cdot)\) \(\chi_{539}(137,\cdot)\) \(\chi_{539}(158,\cdot)\) \(\chi_{539}(163,\cdot)\) \(\chi_{539}(170,\cdot)\) \(\chi_{539}(179,\cdot)\) \(\chi_{539}(191,\cdot)\) \(\chi_{539}(207,\cdot)\) \(\chi_{539}(212,\cdot)\) \(\chi_{539}(235,\cdot)\) \(\chi_{539}(240,\cdot)\) \(\chi_{539}(247,\cdot)\) \(\chi_{539}(256,\cdot)\) \(\chi_{539}(268,\cdot)\) \(\chi_{539}(284,\cdot)\) \(\chi_{539}(289,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{105})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 105 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((199,442)\) → \((e\left(\frac{17}{21}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 539 }(284, a) \) \(1\)\(1\)\(e\left(\frac{68}{105}\right)\)\(e\left(\frac{64}{105}\right)\)\(e\left(\frac{31}{105}\right)\)\(e\left(\frac{92}{105}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{23}{105}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{35}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 539 }(284,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(284,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(284,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(284,·)) \;\) at \(\; a,b = \) e.g. 1,2