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Properties

Label 3547.1045
Modulus $3547$
Conductor $3547$
Order $1773$
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(3547, base_ring=CyclotomicField(3546)) M = H._module chi = DirichletCharacter(H, M([2266]))
 
Copy content gp:[g,chi] = znchar(Mod(1045, 3547))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3547.1045");
 

Basic properties

Modulus: \(3547\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3547\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1773\)
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 3547.k

\(\chi_{3547}(4,\cdot)\) \(\chi_{3547}(9,\cdot)\) \(\chi_{3547}(10,\cdot)\) \(\chi_{3547}(16,\cdot)\) \(\chi_{3547}(19,\cdot)\) \(\chi_{3547}(23,\cdot)\) \(\chi_{3547}(24,\cdot)\) \(\chi_{3547}(25,\cdot)\) \(\chi_{3547}(26,\cdot)\) \(\chi_{3547}(28,\cdot)\) \(\chi_{3547}(29,\cdot)\) \(\chi_{3547}(33,\cdot)\) \(\chi_{3547}(34,\cdot)\) \(\chi_{3547}(40,\cdot)\) \(\chi_{3547}(47,\cdot)\) \(\chi_{3547}(51,\cdot)\) \(\chi_{3547}(53,\cdot)\) \(\chi_{3547}(54,\cdot)\) \(\chi_{3547}(60,\cdot)\) \(\chi_{3547}(61,\cdot)\) \(\chi_{3547}(63,\cdot)\) \(\chi_{3547}(65,\cdot)\) \(\chi_{3547}(67,\cdot)\) \(\chi_{3547}(70,\cdot)\) \(\chi_{3547}(71,\cdot)\) \(\chi_{3547}(74,\cdot)\) \(\chi_{3547}(79,\cdot)\) \(\chi_{3547}(81,\cdot)\) \(\chi_{3547}(85,\cdot)\) \(\chi_{3547}(86,\cdot)\) ...

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: $\Q(\zeta_{1773})$
Fixed field: Number field defined by a degree 1773 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1133}{1773}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3547 }(1045, a) \) \(1\)\(1\)\(e\left(\frac{1133}{1773}\right)\)\(e\left(\frac{592}{1773}\right)\)\(e\left(\frac{493}{1773}\right)\)\(e\left(\frac{1640}{1773}\right)\)\(e\left(\frac{575}{591}\right)\)\(e\left(\frac{127}{197}\right)\)\(e\left(\frac{542}{591}\right)\)\(e\left(\frac{1184}{1773}\right)\)\(e\left(\frac{1000}{1773}\right)\)\(e\left(\frac{430}{1773}\right)\)
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)
 
\( \chi_{ 3547 }(1045,a) \;\) at \(\;a = \) e.g. 2