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Properties

Label 65869.5774
Modulus $65869$
Conductor $65869$
Order $990$
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(65869, base_ring=CyclotomicField(990)) M = H._module chi = DirichletCharacter(H, M([5,489]))
 
Copy content gp:[g,chi] = znchar(Mod(5774, 65869))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("65869.5774");
 

Basic properties

Modulus: \(65869\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(65869\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(990\)
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 65869.bfx

\(\chi_{65869}(532,\cdot)\) \(\chi_{65869}(717,\cdot)\) \(\chi_{65869}(1197,\cdot)\) \(\chi_{65869}(1943,\cdot)\) \(\chi_{65869}(2144,\cdot)\) \(\chi_{65869}(2372,\cdot)\) \(\chi_{65869}(2465,\cdot)\) \(\chi_{65869}(2784,\cdot)\) \(\chi_{65869}(3072,\cdot)\) \(\chi_{65869}(3131,\cdot)\) \(\chi_{65869}(3228,\cdot)\) \(\chi_{65869}(4400,\cdot)\) \(\chi_{65869}(4511,\cdot)\) \(\chi_{65869}(4530,\cdot)\) \(\chi_{65869}(4824,\cdot)\) \(\chi_{65869}(5005,\cdot)\) \(\chi_{65869}(5324,\cdot)\) \(\chi_{65869}(5337,\cdot)\) \(\chi_{65869}(5340,\cdot)\) \(\chi_{65869}(5486,\cdot)\) \(\chi_{65869}(5558,\cdot)\) \(\chi_{65869}(5677,\cdot)\) \(\chi_{65869}(5774,\cdot)\) \(\chi_{65869}(5934,\cdot)\) \(\chi_{65869}(5937,\cdot)\) \(\chi_{65869}(6000,\cdot)\) \(\chi_{65869}(6118,\cdot)\) \(\chi_{65869}(6159,\cdot)\) \(\chi_{65869}(6167,\cdot)\) \(\chi_{65869}(6244,\cdot)\) ...

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: $\Q(\zeta_{495})$
Fixed field: Number field defined by a degree 990 polynomial (not computed)

Values on generators

\((64877,996)\) → \((e\left(\frac{1}{198}\right),e\left(\frac{163}{330}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 65869 }(5774, a) \) \(1\)\(1\)\(e\left(\frac{299}{990}\right)\)\(e\left(\frac{247}{495}\right)\)\(e\left(\frac{299}{495}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{793}{990}\right)\)\(e\left(\frac{719}{990}\right)\)\(e\left(\frac{299}{330}\right)\)\(e\left(\frac{494}{495}\right)\)\(e\left(\frac{563}{990}\right)\)\(e\left(\frac{103}{165}\right)\)
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)
 
\( \chi_{ 65869 }(5774,a) \;\) at \(\;a = \) e.g. 2