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Properties

Label 4013.3
Modulus $4013$
Conductor $4013$
Order $4012$
Real no
Primitive yes
Minimal yes
Parity odd

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(4013, base_ring=CyclotomicField(4012)) M = H._module chi = DirichletCharacter(H, M([1745]))
 
Copy content gp:[g,chi] = znchar(Mod(3, 4013))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4013.3");
 

Basic properties

Modulus: \(4013\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4013\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4012\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 4013.l

\(\chi_{4013}(2,\cdot)\) \(\chi_{4013}(3,\cdot)\) \(\chi_{4013}(5,\cdot)\) \(\chi_{4013}(8,\cdot)\) \(\chi_{4013}(12,\cdot)\) \(\chi_{4013}(14,\cdot)\) \(\chi_{4013}(18,\cdot)\) \(\chi_{4013}(20,\cdot)\) \(\chi_{4013}(21,\cdot)\) \(\chi_{4013}(22,\cdot)\) \(\chi_{4013}(23,\cdot)\) \(\chi_{4013}(26,\cdot)\) \(\chi_{4013}(27,\cdot)\) \(\chi_{4013}(29,\cdot)\) \(\chi_{4013}(30,\cdot)\) \(\chi_{4013}(32,\cdot)\) \(\chi_{4013}(33,\cdot)\) \(\chi_{4013}(34,\cdot)\) \(\chi_{4013}(35,\cdot)\) \(\chi_{4013}(37,\cdot)\) \(\chi_{4013}(38,\cdot)\) \(\chi_{4013}(39,\cdot)\) \(\chi_{4013}(45,\cdot)\) \(\chi_{4013}(48,\cdot)\) \(\chi_{4013}(50,\cdot)\) \(\chi_{4013}(55,\cdot)\) \(\chi_{4013}(56,\cdot)\) \(\chi_{4013}(57,\cdot)\) \(\chi_{4013}(62,\cdot)\) \(\chi_{4013}(65,\cdot)\) ...

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: $\Q(\zeta_{4012})$
Fixed field: Number field defined by a degree 4012 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1745}{4012}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4013 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{1745}{4012}\right)\)\(e\left(\frac{3929}{4012}\right)\)\(e\left(\frac{1745}{2006}\right)\)\(e\left(\frac{2857}{4012}\right)\)\(e\left(\frac{831}{2006}\right)\)\(e\left(\frac{77}{1003}\right)\)\(e\left(\frac{1223}{4012}\right)\)\(e\left(\frac{1923}{2006}\right)\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{703}{1003}\right)\)
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)
 
\( \chi_{ 4013 }(3,a) \;\) at \(\;a = \) e.g. 2