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Properties

Label 11849.300
Modulus $11849$
Conductor $11849$
Order $1360$
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(11849, base_ring=CyclotomicField(1360)) M = H._module chi = DirichletCharacter(H, M([115,1054]))
 
Copy content gp:[g,chi] = znchar(Mod(300, 11849))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11849.300");
 

Basic properties

Modulus: \(11849\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11849\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1360\)
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 11849.ej

\(\chi_{11849}(22,\cdot)\) \(\chi_{11849}(56,\cdot)\) \(\chi_{11849}(63,\cdot)\) \(\chi_{11849}(88,\cdot)\) \(\chi_{11849}(97,\cdot)\) \(\chi_{11849}(99,\cdot)\) \(\chi_{11849}(130,\cdot)\) \(\chi_{11849}(190,\cdot)\) \(\chi_{11849}(193,\cdot)\) \(\chi_{11849}(199,\cdot)\) \(\chi_{11849}(252,\cdot)\) \(\chi_{11849}(299,\cdot)\) \(\chi_{11849}(300,\cdot)\) \(\chi_{11849}(311,\cdot)\) \(\chi_{11849}(317,\cdot)\) \(\chi_{11849}(352,\cdot)\) \(\chi_{11849}(381,\cdot)\) \(\chi_{11849}(388,\cdot)\) \(\chi_{11849}(499,\cdot)\) \(\chi_{11849}(520,\cdot)\) \(\chi_{11849}(521,\cdot)\) \(\chi_{11849}(585,\cdot)\) \(\chi_{11849}(602,\cdot)\) \(\chi_{11849}(632,\cdot)\) \(\chi_{11849}(669,\cdot)\) \(\chi_{11849}(686,\cdot)\) \(\chi_{11849}(690,\cdot)\) \(\chi_{11849}(719,\cdot)\) \(\chi_{11849}(753,\cdot)\) \(\chi_{11849}(760,\cdot)\) ...

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: $\Q(\zeta_{1360})$
Fixed field: Number field defined by a degree 1360 polynomial (not computed)

Values on generators

\((11563,6648)\) → \((e\left(\frac{23}{272}\right),e\left(\frac{31}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11849 }(300, a) \) \(1\)\(1\)\(e\left(\frac{147}{680}\right)\)\(e\left(\frac{193}{272}\right)\)\(e\left(\frac{147}{340}\right)\)\(e\left(\frac{563}{1360}\right)\)\(e\left(\frac{1259}{1360}\right)\)\(e\left(\frac{451}{1360}\right)\)\(e\left(\frac{441}{680}\right)\)\(e\left(\frac{57}{136}\right)\)\(e\left(\frac{857}{1360}\right)\)\(e\left(\frac{367}{1360}\right)\)
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)
 
\( \chi_{ 11849 }(300,a) \;\) at \(\;a = \) e.g. 2