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Properties

Label 4219.97
Modulus $4219$
Conductor $4219$
Order $703$
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(4219, base_ring=CyclotomicField(1406)) M = H._module chi = DirichletCharacter(H, M([506]))
 
Copy content gp:[g,chi] = znchar(Mod(97, 4219))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4219.97");
 

Basic properties

Modulus: \(4219\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4219\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(703\)
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 4219.m

\(\chi_{4219}(5,\cdot)\) \(\chi_{4219}(6,\cdot)\) \(\chi_{4219}(11,\cdot)\) \(\chi_{4219}(23,\cdot)\) \(\chi_{4219}(25,\cdot)\) \(\chi_{4219}(28,\cdot)\) \(\chi_{4219}(30,\cdot)\) \(\chi_{4219}(34,\cdot)\) \(\chi_{4219}(36,\cdot)\) \(\chi_{4219}(37,\cdot)\) \(\chi_{4219}(55,\cdot)\) \(\chi_{4219}(64,\cdot)\) \(\chi_{4219}(66,\cdot)\) \(\chi_{4219}(87,\cdot)\) \(\chi_{4219}(97,\cdot)\) \(\chi_{4219}(104,\cdot)\) \(\chi_{4219}(106,\cdot)\) \(\chi_{4219}(115,\cdot)\) \(\chi_{4219}(121,\cdot)\) \(\chi_{4219}(122,\cdot)\) \(\chi_{4219}(125,\cdot)\) \(\chi_{4219}(134,\cdot)\) \(\chi_{4219}(137,\cdot)\) \(\chi_{4219}(140,\cdot)\) \(\chi_{4219}(141,\cdot)\) \(\chi_{4219}(150,\cdot)\) \(\chi_{4219}(152,\cdot)\) \(\chi_{4219}(155,\cdot)\) \(\chi_{4219}(164,\cdot)\) \(\chi_{4219}(168,\cdot)\) ...

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: $\Q(\zeta_{703})$
Fixed field: Number field defined by a degree 703 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{253}{703}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4219 }(97, a) \) \(1\)\(1\)\(e\left(\frac{253}{703}\right)\)\(e\left(\frac{55}{703}\right)\)\(e\left(\frac{506}{703}\right)\)\(e\left(\frac{486}{703}\right)\)\(e\left(\frac{308}{703}\right)\)\(e\left(\frac{394}{703}\right)\)\(e\left(\frac{56}{703}\right)\)\(e\left(\frac{110}{703}\right)\)\(e\left(\frac{36}{703}\right)\)\(e\left(\frac{66}{703}\right)\)
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)
 
\( \chi_{ 4219 }(97,a) \;\) at \(\;a = \) e.g. 2