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Properties

Label 4183.117
Modulus $4183$
Conductor $4183$
Order $2024$
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(4183, base_ring=CyclotomicField(2024)) M = H._module chi = DirichletCharacter(H, M([220,575]))
 
Copy content gp:[g,chi] = znchar(Mod(117, 4183))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4183.117");
 

Basic properties

Modulus: \(4183\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4183\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2024\)
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 4183.bf

\(\chi_{4183}(13,\cdot)\) \(\chi_{4183}(15,\cdot)\) \(\chi_{4183}(19,\cdot)\) \(\chi_{4183}(23,\cdot)\) \(\chi_{4183}(26,\cdot)\) \(\chi_{4183}(29,\cdot)\) \(\chi_{4183}(30,\cdot)\) \(\chi_{4183}(31,\cdot)\) \(\chi_{4183}(33,\cdot)\) \(\chi_{4183}(35,\cdot)\) \(\chi_{4183}(38,\cdot)\) \(\chi_{4183}(41,\cdot)\) \(\chi_{4183}(43,\cdot)\) \(\chi_{4183}(58,\cdot)\) \(\chi_{4183}(60,\cdot)\) \(\chi_{4183}(62,\cdot)\) \(\chi_{4183}(66,\cdot)\) \(\chi_{4183}(70,\cdot)\) \(\chi_{4183}(76,\cdot)\) \(\chi_{4183}(82,\cdot)\) \(\chi_{4183}(86,\cdot)\) \(\chi_{4183}(92,\cdot)\) \(\chi_{4183}(104,\cdot)\) \(\chi_{4183}(113,\cdot)\) \(\chi_{4183}(116,\cdot)\) \(\chi_{4183}(117,\cdot)\) \(\chi_{4183}(120,\cdot)\) \(\chi_{4183}(124,\cdot)\) \(\chi_{4183}(127,\cdot)\) \(\chi_{4183}(132,\cdot)\) ...

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: $\Q(\zeta_{2024})$
Fixed field: Number field defined by a degree 2024 polynomial (not computed)

Values on generators

\((1603,3385)\) → \((e\left(\frac{5}{46}\right),e\left(\frac{25}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4183 }(117, a) \) \(1\)\(1\)\(e\left(\frac{127}{253}\right)\)\(e\left(\frac{927}{2024}\right)\)\(e\left(\frac{1}{253}\right)\)\(e\left(\frac{1007}{1012}\right)\)\(e\left(\frac{1943}{2024}\right)\)\(e\left(\frac{991}{2024}\right)\)\(e\left(\frac{128}{253}\right)\)\(e\left(\frac{927}{1012}\right)\)\(e\left(\frac{503}{1012}\right)\)\(e\left(\frac{158}{253}\right)\)
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)
 
\( \chi_{ 4183 }(117,a) \;\) at \(\;a = \) e.g. 2