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Properties

Label 947.914
Modulus $947$
Conductor $947$
Order $43$
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(947, base_ring=CyclotomicField(86)) M = H._module chi = DirichletCharacter(H, M([68]))
 
Copy content gp:[g,chi] = znchar(Mod(914, 947))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("947.914");
 

Basic properties

Modulus: \(947\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(947\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(43\)
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 947.e

\(\chi_{947}(22,\cdot)\) \(\chi_{947}(30,\cdot)\) \(\chi_{947}(41,\cdot)\) \(\chi_{947}(49,\cdot)\) \(\chi_{947}(58,\cdot)\) \(\chi_{947}(115,\cdot)\) \(\chi_{947}(127,\cdot)\) \(\chi_{947}(131,\cdot)\) \(\chi_{947}(140,\cdot)\) \(\chi_{947}(142,\cdot)\) \(\chi_{947}(221,\cdot)\) \(\chi_{947}(231,\cdot)\) \(\chi_{947}(239,\cdot)\) \(\chi_{947}(277,\cdot)\) \(\chi_{947}(283,\cdot)\) \(\chi_{947}(301,\cdot)\) \(\chi_{947}(315,\cdot)\) \(\chi_{947}(329,\cdot)\) \(\chi_{947}(347,\cdot)\) \(\chi_{947}(400,\cdot)\) \(\chi_{947}(412,\cdot)\) \(\chi_{947}(472,\cdot)\) \(\chi_{947}(484,\cdot)\) \(\chi_{947}(507,\cdot)\) \(\chi_{947}(523,\cdot)\) \(\chi_{947}(538,\cdot)\) \(\chi_{947}(541,\cdot)\) \(\chi_{947}(544,\cdot)\) \(\chi_{947}(604,\cdot)\) \(\chi_{947}(609,\cdot)\) ...

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: $\Q(\zeta_{43})$
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 43 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(2\) → \(e\left(\frac{34}{43}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 947 }(914, a) \) \(1\)\(1\)\(e\left(\frac{34}{43}\right)\)\(e\left(\frac{30}{43}\right)\)\(e\left(\frac{25}{43}\right)\)\(e\left(\frac{23}{43}\right)\)\(e\left(\frac{21}{43}\right)\)\(e\left(\frac{7}{43}\right)\)\(e\left(\frac{16}{43}\right)\)\(e\left(\frac{17}{43}\right)\)\(e\left(\frac{14}{43}\right)\)\(e\left(\frac{32}{43}\right)\)
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)
 
\( \chi_{ 947 }(914,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 947 }(914,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 947 }(914,·),\chi_{ 947 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 947 }(914,·)) \;\) at \(\; a,b = \) e.g. 1,2