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Properties

Label 7547.36
Modulus $7547$
Conductor $7547$
Order $3773$
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(7547, base_ring=CyclotomicField(7546)) M = H._module chi = DirichletCharacter(H, M([5052]))
 
Copy content gp:[g,chi] = znchar(Mod(36, 7547))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7547.36");
 

Basic properties

Modulus: \(7547\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7547\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3773\)
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 7547.o

\(\chi_{7547}(3,\cdot)\) \(\chi_{7547}(4,\cdot)\) \(\chi_{7547}(9,\cdot)\) \(\chi_{7547}(10,\cdot)\) \(\chi_{7547}(14,\cdot)\) \(\chi_{7547}(16,\cdot)\) \(\chi_{7547}(22,\cdot)\) \(\chi_{7547}(25,\cdot)\) \(\chi_{7547}(26,\cdot)\) \(\chi_{7547}(27,\cdot)\) \(\chi_{7547}(29,\cdot)\) \(\chi_{7547}(30,\cdot)\) \(\chi_{7547}(35,\cdot)\) \(\chi_{7547}(36,\cdot)\) \(\chi_{7547}(38,\cdot)\) \(\chi_{7547}(40,\cdot)\) \(\chi_{7547}(43,\cdot)\) \(\chi_{7547}(46,\cdot)\) \(\chi_{7547}(48,\cdot)\) \(\chi_{7547}(49,\cdot)\) \(\chi_{7547}(51,\cdot)\) \(\chi_{7547}(55,\cdot)\) \(\chi_{7547}(56,\cdot)\) \(\chi_{7547}(59,\cdot)\) \(\chi_{7547}(62,\cdot)\) \(\chi_{7547}(64,\cdot)\) \(\chi_{7547}(65,\cdot)\) \(\chi_{7547}(66,\cdot)\) \(\chi_{7547}(67,\cdot)\) \(\chi_{7547}(68,\cdot)\) ...

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: $\Q(\zeta_{3773})$
Fixed field: Number field defined by a degree 3773 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2526}{3773}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 7547 }(36, a) \) \(1\)\(1\)\(e\left(\frac{2526}{3773}\right)\)\(e\left(\frac{1780}{3773}\right)\)\(e\left(\frac{1279}{3773}\right)\)\(e\left(\frac{194}{3773}\right)\)\(e\left(\frac{533}{3773}\right)\)\(e\left(\frac{3506}{3773}\right)\)\(e\left(\frac{32}{3773}\right)\)\(e\left(\frac{3560}{3773}\right)\)\(e\left(\frac{2720}{3773}\right)\)\(e\left(\frac{2048}{3773}\right)\)
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)
 
\( \chi_{ 7547 }(36,a) \;\) at \(\;a = \) e.g. 2