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Properties

Label 743.212
Modulus $743$
Conductor $743$
Order $53$
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(743, base_ring=CyclotomicField(106)) M = H._module chi = DirichletCharacter(H, M([2]))
 
Copy content gp:[g,chi] = znchar(Mod(212, 743))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("743.212");
 

Basic properties

Modulus: \(743\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(743\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(53\)
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 743.e

\(\chi_{743}(12,\cdot)\) \(\chi_{743}(37,\cdot)\) \(\chi_{743}(38,\cdot)\) \(\chi_{743}(50,\cdot)\) \(\chi_{743}(62,\cdot)\) \(\chi_{743}(65,\cdot)\) \(\chi_{743}(82,\cdot)\) \(\chi_{743}(94,\cdot)\) \(\chi_{743}(127,\cdot)\) \(\chi_{743}(128,\cdot)\) \(\chi_{743}(129,\cdot)\) \(\chi_{743}(144,\cdot)\) \(\chi_{743}(147,\cdot)\) \(\chi_{743}(162,\cdot)\) \(\chi_{743}(166,\cdot)\) \(\chi_{743}(176,\cdot)\) \(\chi_{743}(198,\cdot)\) \(\chi_{743}(212,\cdot)\) \(\chi_{743}(238,\cdot)\) \(\chi_{743}(239,\cdot)\) \(\chi_{743}(241,\cdot)\) \(\chi_{743}(242,\cdot)\) \(\chi_{743}(271,\cdot)\) \(\chi_{743}(278,\cdot)\) \(\chi_{743}(280,\cdot)\) \(\chi_{743}(295,\cdot)\) \(\chi_{743}(315,\cdot)\) \(\chi_{743}(364,\cdot)\) \(\chi_{743}(368,\cdot)\) \(\chi_{743}(385,\cdot)\) ...

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: $\Q(\zeta_{53})$
Fixed field: Number field defined by a degree 53 polynomial

Values on generators

\(5\) → \(e\left(\frac{1}{53}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 743 }(212, a) \) \(1\)\(1\)\(e\left(\frac{9}{53}\right)\)\(e\left(\frac{36}{53}\right)\)\(e\left(\frac{18}{53}\right)\)\(e\left(\frac{1}{53}\right)\)\(e\left(\frac{45}{53}\right)\)\(e\left(\frac{48}{53}\right)\)\(e\left(\frac{27}{53}\right)\)\(e\left(\frac{19}{53}\right)\)\(e\left(\frac{10}{53}\right)\)\(e\left(\frac{50}{53}\right)\)
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)
 
\( \chi_{ 743 }(212,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 743 }(212,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 743 }(212,·),\chi_{ 743 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 743 }(212,·)) \;\) at \(\; a,b = \) e.g. 1,2