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Properties

Label 4069.1746
Modulus $4069$
Conductor $4069$
Order $78$
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(4069, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([13,71]))
 
Copy content gp:[g,chi] = znchar(Mod(1746, 4069))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4069.1746");
 

Basic properties

Modulus: \(4069\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4069\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 4069.dr

\(\chi_{4069}(4,\cdot)\) \(\chi_{4069}(108,\cdot)\) \(\chi_{4069}(543,\cdot)\) \(\chi_{4069}(550,\cdot)\) \(\chi_{4069}(1024,\cdot)\) \(\chi_{4069}(1226,\cdot)\) \(\chi_{4069}(1421,\cdot)\) \(\chi_{4069}(1635,\cdot)\) \(\chi_{4069}(1746,\cdot)\) \(\chi_{4069}(1869,\cdot)\) \(\chi_{4069}(2110,\cdot)\) \(\chi_{4069}(2188,\cdot)\) \(\chi_{4069}(2383,\cdot)\) \(\chi_{4069}(2454,\cdot)\) \(\chi_{4069}(2643,\cdot)\) \(\chi_{4069}(2675,\cdot)\) \(\chi_{4069}(3052,\cdot)\) \(\chi_{4069}(3234,\cdot)\) \(\chi_{4069}(3306,\cdot)\) \(\chi_{4069}(3455,\cdot)\) \(\chi_{4069}(3637,\cdot)\) \(\chi_{4069}(3767,\cdot)\) \(\chi_{4069}(3813,\cdot)\) \(\chi_{4069}(4053,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((314,1262)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{71}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4069 }(1746, a) \) \(1\)\(1\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(1\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{23}{26}\right)\)
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)
 
\( \chi_{ 4069 }(1746,a) \;\) at \(\;a = \) e.g. 2