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Properties

Label 106069.6067
Modulus $106069$
Conductor $106069$
Order $1452$
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(106069, base_ring=CyclotomicField(1452)) M = H._module chi = DirichletCharacter(H, M([484,1435]))
 
Copy content gp:[g,chi] = znchar(Mod(6067, 106069))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("106069.6067");
 

Basic properties

Modulus: \(106069\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(106069\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1452\)
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 106069.kp

\(\chi_{106069}(227,\cdot)\) \(\chi_{106069}(648,\cdot)\) \(\chi_{106069}(811,\cdot)\) \(\chi_{106069}(1541,\cdot)\) \(\chi_{106069}(1833,\cdot)\) \(\chi_{106069}(1979,\cdot)\) \(\chi_{106069}(2125,\cdot)\) \(\chi_{106069}(2198,\cdot)\) \(\chi_{106069}(2692,\cdot)\) \(\chi_{106069}(3130,\cdot)\) \(\chi_{106069}(3220,\cdot)\) \(\chi_{106069}(3366,\cdot)\) \(\chi_{106069}(3495,\cdot)\) \(\chi_{106069}(4298,\cdot)\) \(\chi_{106069}(4444,\cdot)\) \(\chi_{106069}(4461,\cdot)\) \(\chi_{106069}(4882,\cdot)\) \(\chi_{106069}(5045,\cdot)\) \(\chi_{106069}(5483,\cdot)\) \(\chi_{106069}(5612,\cdot)\) \(\chi_{106069}(5629,\cdot)\) \(\chi_{106069}(5685,\cdot)\) \(\chi_{106069}(6050,\cdot)\) \(\chi_{106069}(6067,\cdot)\) \(\chi_{106069}(6140,\cdot)\) \(\chi_{106069}(6359,\cdot)\) \(\chi_{106069}(6578,\cdot)\) \(\chi_{106069}(6780,\cdot)\) \(\chi_{106069}(6853,\cdot)\) \(\chi_{106069}(6999,\cdot)\) ...

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: $\Q(\zeta_{1452})$
Fixed field: Number field defined by a degree 1452 polynomial (not computed)

Values on generators

\((90087,30515)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1435}{1452}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 106069 }(6067, a) \) \(-1\)\(1\)\(e\left(\frac{317}{484}\right)\)\(e\left(\frac{179}{363}\right)\)\(e\left(\frac{75}{242}\right)\)\(e\left(\frac{37}{132}\right)\)\(e\left(\frac{215}{1452}\right)\)\(e\left(\frac{20}{121}\right)\)\(e\left(\frac{467}{484}\right)\)\(e\left(\frac{358}{363}\right)\)\(e\left(\frac{679}{726}\right)\)\(e\left(\frac{28}{33}\right)\)
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)
 
\( \chi_{ 106069 }(6067,a) \;\) at \(\;a = \) e.g. 2