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Properties

Label 1673.11
Modulus $1673$
Conductor $1673$
Order $357$
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(1673, base_ring=CyclotomicField(714)) M = H._module chi = DirichletCharacter(H, M([476,12]))
 
Copy content gp:[g,chi] = znchar(Mod(11, 1673))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1673.11");
 

Basic properties

Modulus: \(1673\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1673\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(357\)
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 1673.bc

\(\chi_{1673}(2,\cdot)\) \(\chi_{1673}(4,\cdot)\) \(\chi_{1673}(9,\cdot)\) \(\chi_{1673}(11,\cdot)\) \(\chi_{1673}(16,\cdot)\) \(\chi_{1673}(18,\cdot)\) \(\chi_{1673}(25,\cdot)\) \(\chi_{1673}(30,\cdot)\) \(\chi_{1673}(32,\cdot)\) \(\chi_{1673}(58,\cdot)\) \(\chi_{1673}(60,\cdot)\) \(\chi_{1673}(72,\cdot)\) \(\chi_{1673}(81,\cdot)\) \(\chi_{1673}(88,\cdot)\) \(\chi_{1673}(93,\cdot)\) \(\chi_{1673}(102,\cdot)\) \(\chi_{1673}(109,\cdot)\) \(\chi_{1673}(116,\cdot)\) \(\chi_{1673}(121,\cdot)\) \(\chi_{1673}(135,\cdot)\) \(\chi_{1673}(142,\cdot)\) \(\chi_{1673}(144,\cdot)\) \(\chi_{1673}(165,\cdot)\) \(\chi_{1673}(170,\cdot)\) \(\chi_{1673}(186,\cdot)\) \(\chi_{1673}(193,\cdot)\) \(\chi_{1673}(198,\cdot)\) \(\chi_{1673}(200,\cdot)\) \(\chi_{1673}(226,\cdot)\) \(\chi_{1673}(242,\cdot)\) ...

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: $\Q(\zeta_{357})$
Fixed field: Number field defined by a degree 357 polynomial (not computed)

Values on generators

\((479,246)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{119}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1673 }(11, a) \) \(1\)\(1\)\(e\left(\frac{158}{357}\right)\)\(e\left(\frac{325}{357}\right)\)\(e\left(\frac{316}{357}\right)\)\(e\left(\frac{233}{357}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{39}{119}\right)\)\(e\left(\frac{293}{357}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{262}{357}\right)\)\(e\left(\frac{284}{357}\right)\)
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)
 
\( \chi_{ 1673 }(11,a) \;\) at \(\;a = \) e.g. 2