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Properties

Label 1453.263
Modulus $1453$
Conductor $1453$
Order $242$
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(1453, base_ring=CyclotomicField(242)) M = H._module chi = DirichletCharacter(H, M([123]))
 
Copy content gp:[g,chi] = znchar(Mod(263, 1453))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1453.263");
 

Basic properties

Modulus: \(1453\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1453\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(242\)
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 1453.n

\(\chi_{1453}(36,\cdot)\) \(\chi_{1453}(44,\cdot)\) \(\chi_{1453}(51,\cdot)\) \(\chi_{1453}(52,\cdot)\) \(\chi_{1453}(57,\cdot)\) \(\chi_{1453}(64,\cdot)\) \(\chi_{1453}(90,\cdot)\) \(\chi_{1453}(92,\cdot)\) \(\chi_{1453}(110,\cdot)\) \(\chi_{1453}(123,\cdot)\) \(\chi_{1453}(130,\cdot)\) \(\chi_{1453}(149,\cdot)\) \(\chi_{1453}(157,\cdot)\) \(\chi_{1453}(160,\cdot)\) \(\chi_{1453}(166,\cdot)\) \(\chi_{1453}(202,\cdot)\) \(\chi_{1453}(215,\cdot)\) \(\chi_{1453}(225,\cdot)\) \(\chi_{1453}(230,\cdot)\) \(\chi_{1453}(252,\cdot)\) \(\chi_{1453}(254,\cdot)\) \(\chi_{1453}(263,\cdot)\) \(\chi_{1453}(271,\cdot)\) \(\chi_{1453}(305,\cdot)\) \(\chi_{1453}(308,\cdot)\) \(\chi_{1453}(311,\cdot)\) \(\chi_{1453}(325,\cdot)\) \(\chi_{1453}(357,\cdot)\) \(\chi_{1453}(364,\cdot)\) \(\chi_{1453}(388,\cdot)\) ...

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: $\Q(\zeta_{121})$
Fixed field: Number field defined by a degree 242 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{123}{242}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1453 }(263, a) \) \(1\)\(1\)\(e\left(\frac{123}{242}\right)\)\(e\left(\frac{86}{121}\right)\)\(e\left(\frac{2}{121}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{53}{242}\right)\)\(e\left(\frac{114}{121}\right)\)\(e\left(\frac{127}{242}\right)\)\(e\left(\frac{51}{121}\right)\)\(e\left(\frac{34}{121}\right)\)\(e\left(\frac{7}{11}\right)\)
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)
 
\( \chi_{ 1453 }(263,a) \;\) at \(\;a = \) e.g. 2