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Properties

Label 137.105
Modulus $137$
Conductor $137$
Order $68$
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(137, base_ring=CyclotomicField(68)) M = H._module chi = DirichletCharacter(H, M([59]))
 
Copy content gp:[g,chi] = znchar(Mod(105, 137))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("137.105");
 

Basic properties

Modulus: \(137\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(137\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(68\)
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 137.g

\(\chi_{137}(2,\cdot)\) \(\chi_{137}(7,\cdot)\) \(\chi_{137}(8,\cdot)\) \(\chi_{137}(9,\cdot)\) \(\chi_{137}(11,\cdot)\) \(\chi_{137}(17,\cdot)\) \(\chi_{137}(19,\cdot)\) \(\chi_{137}(25,\cdot)\) \(\chi_{137}(28,\cdot)\) \(\chi_{137}(30,\cdot)\) \(\chi_{137}(32,\cdot)\) \(\chi_{137}(36,\cdot)\) \(\chi_{137}(39,\cdot)\) \(\chi_{137}(44,\cdot)\) \(\chi_{137}(61,\cdot)\) \(\chi_{137}(68,\cdot)\) \(\chi_{137}(69,\cdot)\) \(\chi_{137}(76,\cdot)\) \(\chi_{137}(93,\cdot)\) \(\chi_{137}(98,\cdot)\) \(\chi_{137}(101,\cdot)\) \(\chi_{137}(105,\cdot)\) \(\chi_{137}(107,\cdot)\) \(\chi_{137}(109,\cdot)\) \(\chi_{137}(112,\cdot)\) \(\chi_{137}(118,\cdot)\) \(\chi_{137}(120,\cdot)\) \(\chi_{137}(126,\cdot)\) \(\chi_{137}(128,\cdot)\) \(\chi_{137}(129,\cdot)\) ...

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: $\Q(\zeta_{68})$
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 68 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(3\) → \(e\left(\frac{59}{68}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 137 }(105, a) \) \(1\)\(1\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{59}{68}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{5}{68}\right)\)\(e\left(\frac{37}{68}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{25}{34}\right)\)\(-i\)\(e\left(\frac{29}{34}\right)\)
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)
 
\( \chi_{ 137 }(105,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 137 }(105,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(105,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(105,·)) \;\) at \(\; a,b = \) e.g. 1,2