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Properties

Label 547.115
Modulus $547$
Conductor $547$
Order $273$
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(547, base_ring=CyclotomicField(546)) M = H._module chi = DirichletCharacter(H, M([472]))
 
Copy content gp:[g,chi] = znchar(Mod(115, 547))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("547.115");
 

Basic properties

Modulus: \(547\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(547\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(273\)
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 547.o

\(\chi_{547}(4,\cdot)\) \(\chi_{547}(6,\cdot)\) \(\chi_{547}(15,\cdot)\) \(\chi_{547}(16,\cdot)\) \(\chi_{547}(19,\cdot)\) \(\chi_{547}(25,\cdot)\) \(\chi_{547}(34,\cdot)\) \(\chi_{547}(36,\cdot)\) \(\chi_{547}(49,\cdot)\) \(\chi_{547}(51,\cdot)\) \(\chi_{547}(53,\cdot)\) \(\chi_{547}(56,\cdot)\) \(\chi_{547}(60,\cdot)\) \(\chi_{547}(62,\cdot)\) \(\chi_{547}(66,\cdot)\) \(\chi_{547}(67,\cdot)\) \(\chi_{547}(69,\cdot)\) \(\chi_{547}(73,\cdot)\) \(\chi_{547}(74,\cdot)\) \(\chi_{547}(76,\cdot)\) \(\chi_{547}(78,\cdot)\) \(\chi_{547}(82,\cdot)\) \(\chi_{547}(86,\cdot)\) \(\chi_{547}(97,\cdot)\) \(\chi_{547}(99,\cdot)\) \(\chi_{547}(110,\cdot)\) \(\chi_{547}(111,\cdot)\) \(\chi_{547}(113,\cdot)\) \(\chi_{547}(115,\cdot)\) \(\chi_{547}(116,\cdot)\) ...

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: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 273 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{236}{273}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 547 }(115, a) \) \(1\)\(1\)\(e\left(\frac{236}{273}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{199}{273}\right)\)\(e\left(\frac{202}{273}\right)\)\(e\left(\frac{197}{273}\right)\)\(e\left(\frac{11}{273}\right)\)\(e\left(\frac{54}{91}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{55}{91}\right)\)\(e\left(\frac{20}{39}\right)\)
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)
 
\( \chi_{ 547 }(115,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 547 }(115,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(115,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(115,·)) \;\) at \(\; a,b = \) e.g. 1,2