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Properties

Label 689.4
Modulus $689$
Conductor $689$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(689, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([13,3]))
 
Copy content gp:[g,chi] = znchar(Mod(4, 689))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("689.4");
 

Basic properties

Modulus: \(689\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(689\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 689.bh

\(\chi_{689}(4,\cdot)\) \(\chi_{689}(17,\cdot)\) \(\chi_{689}(43,\cdot)\) \(\chi_{689}(62,\cdot)\) \(\chi_{689}(82,\cdot)\) \(\chi_{689}(166,\cdot)\) \(\chi_{689}(199,\cdot)\) \(\chi_{689}(218,\cdot)\) \(\chi_{689}(290,\cdot)\) \(\chi_{689}(303,\cdot)\) \(\chi_{689}(322,\cdot)\) \(\chi_{689}(329,\cdot)\) \(\chi_{689}(335,\cdot)\) \(\chi_{689}(355,\cdot)\) \(\chi_{689}(361,\cdot)\) \(\chi_{689}(400,\cdot)\) \(\chi_{689}(433,\cdot)\) \(\chi_{689}(517,\cdot)\) \(\chi_{689}(537,\cdot)\) \(\chi_{689}(589,\cdot)\) \(\chi_{689}(608,\cdot)\) \(\chi_{689}(621,\cdot)\) \(\chi_{689}(647,\cdot)\) \(\chi_{689}(673,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((54,638)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 689 }(4, a) \) \(1\)\(1\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{31}{78}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 689 }(4,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 689 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 689 }(4,·),\chi_{ 689 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 689 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2