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Properties

Label 8909.196
Modulus $8909$
Conductor $8909$
Order $2175$
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(8909, base_ring=CyclotomicField(4350)) M = H._module chi = DirichletCharacter(H, M([2850,3596]))
 
Copy content gp:[g,chi] = znchar(Mod(196, 8909))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8909.196");
 

Basic properties

Modulus: \(8909\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8909\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2175\)
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 8909.bs

\(\chi_{8909}(5,\cdot)\) \(\chi_{8909}(17,\cdot)\) \(\chi_{8909}(21,\cdot)\) \(\chi_{8909}(22,\cdot)\) \(\chi_{8909}(25,\cdot)\) \(\chi_{8909}(36,\cdot)\) \(\chi_{8909}(45,\cdot)\) \(\chi_{8909}(49,\cdot)\) \(\chi_{8909}(62,\cdot)\) \(\chi_{8909}(74,\cdot)\) \(\chi_{8909}(80,\cdot)\) \(\chi_{8909}(88,\cdot)\) \(\chi_{8909}(95,\cdot)\) \(\chi_{8909}(100,\cdot)\) \(\chi_{8909}(116,\cdot)\) \(\chi_{8909}(121,\cdot)\) \(\chi_{8909}(137,\cdot)\) \(\chi_{8909}(138,\cdot)\) \(\chi_{8909}(139,\cdot)\) \(\chi_{8909}(144,\cdot)\) \(\chi_{8909}(145,\cdot)\) \(\chi_{8909}(169,\cdot)\) \(\chi_{8909}(182,\cdot)\) \(\chi_{8909}(193,\cdot)\) \(\chi_{8909}(194,\cdot)\) \(\chi_{8909}(196,\cdot)\) \(\chi_{8909}(198,\cdot)\) \(\chi_{8909}(206,\cdot)\) \(\chi_{8909}(213,\cdot)\) \(\chi_{8909}(225,\cdot)\) ...

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: $\Q(\zeta_{2175})$
Fixed field: Number field defined by a degree 2175 polynomial (not computed)

Values on generators

\((5135,1063)\) → \((e\left(\frac{19}{29}\right),e\left(\frac{62}{75}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8909 }(196, a) \) \(1\)\(1\)\(e\left(\frac{227}{435}\right)\)\(e\left(\frac{521}{725}\right)\)\(e\left(\frac{19}{435}\right)\)\(e\left(\frac{1126}{2175}\right)\)\(e\left(\frac{523}{2175}\right)\)\(e\left(\frac{391}{2175}\right)\)\(e\left(\frac{82}{145}\right)\)\(e\left(\frac{317}{725}\right)\)\(e\left(\frac{86}{2175}\right)\)\(e\left(\frac{187}{2175}\right)\)
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)
 
\( \chi_{ 8909 }(196,a) \;\) at \(\;a = \) e.g. 2