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Properties

Label 559.306
Modulus $559$
Conductor $559$
Order $84$
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(559, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([77,50]))
 
Copy content gp:[g,chi] = znchar(Mod(306, 559))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("559.306");
 

Basic properties

Modulus: \(559\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(559\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 559.cc

\(\chi_{559}(19,\cdot)\) \(\chi_{559}(20,\cdot)\) \(\chi_{559}(28,\cdot)\) \(\chi_{559}(33,\cdot)\) \(\chi_{559}(46,\cdot)\) \(\chi_{559}(98,\cdot)\) \(\chi_{559}(115,\cdot)\) \(\chi_{559}(149,\cdot)\) \(\chi_{559}(158,\cdot)\) \(\chi_{559}(162,\cdot)\) \(\chi_{559}(175,\cdot)\) \(\chi_{559}(206,\cdot)\) \(\chi_{559}(227,\cdot)\) \(\chi_{559}(245,\cdot)\) \(\chi_{559}(249,\cdot)\) \(\chi_{559}(284,\cdot)\) \(\chi_{559}(288,\cdot)\) \(\chi_{559}(306,\cdot)\) \(\chi_{559}(327,\cdot)\) \(\chi_{559}(362,\cdot)\) \(\chi_{559}(405,\cdot)\) \(\chi_{559}(435,\cdot)\) \(\chi_{559}(449,\cdot)\) \(\chi_{559}(544,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((431,261)\) → \((e\left(\frac{11}{12}\right),e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 559 }(306, a) \) \(1\)\(1\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{11}{84}\right)\)\(i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{23}{84}\right)\)
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)
 
\( \chi_{ 559 }(306,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 559 }(306,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 559 }(306,·),\chi_{ 559 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 559 }(306,·)) \;\) at \(\; a,b = \) e.g. 1,2