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Properties

Label 4163.157
Modulus $4163$
Conductor $4163$
Order $1980$
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(4163, base_ring=CyclotomicField(1980)) M = H._module chi = DirichletCharacter(H, M([1350,1639]))
 
Copy content gp:[g,chi] = znchar(Mod(157, 4163))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4163.157");
 

Basic properties

Modulus: \(4163\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4163\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1980\)
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 4163.ct

\(\chi_{4163}(10,\cdot)\) \(\chi_{4163}(21,\cdot)\) \(\chi_{4163}(28,\cdot)\) \(\chi_{4163}(53,\cdot)\) \(\chi_{4163}(57,\cdot)\) \(\chi_{4163}(63,\cdot)\) \(\chi_{4163}(66,\cdot)\) \(\chi_{4163}(76,\cdot)\) \(\chi_{4163}(83,\cdot)\) \(\chi_{4163}(84,\cdot)\) \(\chi_{4163}(90,\cdot)\) \(\chi_{4163}(97,\cdot)\) \(\chi_{4163}(103,\cdot)\) \(\chi_{4163}(112,\cdot)\) \(\chi_{4163}(134,\cdot)\) \(\chi_{4163}(153,\cdot)\) \(\chi_{4163}(157,\cdot)\) \(\chi_{4163}(158,\cdot)\) \(\chi_{4163}(171,\cdot)\) \(\chi_{4163}(191,\cdot)\) \(\chi_{4163}(199,\cdot)\) \(\chi_{4163}(204,\cdot)\) \(\chi_{4163}(205,\cdot)\) \(\chi_{4163}(222,\cdot)\) \(\chi_{4163}(228,\cdot)\) \(\chi_{4163}(235,\cdot)\) \(\chi_{4163}(244,\cdot)\) \(\chi_{4163}(247,\cdot)\) \(\chi_{4163}(250,\cdot)\) \(\chi_{4163}(258,\cdot)\) ...

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: $\Q(\zeta_{1980})$
Fixed field: Number field defined by a degree 1980 polynomial (not computed)

Values on generators

\((2535,1450)\) → \((e\left(\frac{15}{22}\right),e\left(\frac{149}{180}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4163 }(157, a) \) \(1\)\(1\)\(e\left(\frac{379}{1980}\right)\)\(e\left(\frac{131}{495}\right)\)\(e\left(\frac{379}{990}\right)\)\(e\left(\frac{269}{330}\right)\)\(e\left(\frac{301}{660}\right)\)\(e\left(\frac{49}{132}\right)\)\(e\left(\frac{379}{660}\right)\)\(e\left(\frac{262}{495}\right)\)\(e\left(\frac{13}{1980}\right)\)\(e\left(\frac{227}{495}\right)\)
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)
 
\( \chi_{ 4163 }(157,a) \;\) at \(\;a = \) e.g. 2