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Properties

Label 53371.25
Modulus $53371$
Conductor $53371$
Order $12402$
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(53371, base_ring=CyclotomicField(12402)) M = H._module chi = DirichletCharacter(H, M([9646,2061]))
 
Copy content gp:[g,chi] = znchar(Mod(25, 53371))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("53371.25");
 

Basic properties

Modulus: \(53371\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(53371\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(12402\)
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 53371.cq

\(\chi_{53371}(4,\cdot)\) \(\chi_{53371}(6,\cdot)\) \(\chi_{53371}(9,\cdot)\) \(\chi_{53371}(17,\cdot)\) \(\chi_{53371}(25,\cdot)\) \(\chi_{53371}(43,\cdot)\) \(\chi_{53371}(62,\cdot)\) \(\chi_{53371}(82,\cdot)\) \(\chi_{53371}(93,\cdot)\) \(\chi_{53371}(112,\cdot)\) \(\chi_{53371}(123,\cdot)\) \(\chi_{53371}(131,\cdot)\) \(\chi_{53371}(149,\cdot)\) \(\chi_{53371}(168,\cdot)\) \(\chi_{53371}(176,\cdot)\) \(\chi_{53371}(188,\cdot)\) \(\chi_{53371}(196,\cdot)\) \(\chi_{53371}(199,\cdot)\) \(\chi_{53371}(218,\cdot)\) \(\chi_{53371}(237,\cdot)\) \(\chi_{53371}(252,\cdot)\) \(\chi_{53371}(271,\cdot)\) \(\chi_{53371}(272,\cdot)\) \(\chi_{53371}(282,\cdot)\) \(\chi_{53371}(290,\cdot)\) \(\chi_{53371}(294,\cdot)\) \(\chi_{53371}(302,\cdot)\) \(\chi_{53371}(308,\cdot)\) \(\chi_{53371}(327,\cdot)\) \(\chi_{53371}(329,\cdot)\) ...

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: $\Q(\zeta_{6201})$
Fixed field: Number field defined by a degree 12402 polynomial (not computed)

Values on generators

\((16855,36519)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{229}{1378}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 53371 }(25, a) \) \(1\)\(1\)\(e\left(\frac{11707}{12402}\right)\)\(e\left(\frac{8803}{12402}\right)\)\(e\left(\frac{5506}{6201}\right)\)\(e\left(\frac{6205}{12402}\right)\)\(e\left(\frac{4054}{6201}\right)\)\(e\left(\frac{181}{2067}\right)\)\(e\left(\frac{3439}{4134}\right)\)\(e\left(\frac{2602}{6201}\right)\)\(e\left(\frac{2755}{6201}\right)\)\(e\left(\frac{878}{2067}\right)\)
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)
 
\( \chi_{ 53371 }(25,a) \;\) at \(\;a = \) e.g. 2