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Properties

Label 11347.1062
Modulus $11347$
Conductor $11347$
Order $324$
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(11347, base_ring=CyclotomicField(324)) M = H._module chi = DirichletCharacter(H, M([270,145]))
 
Copy content gp:[g,chi] = znchar(Mod(1062, 11347))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11347.1062");
 

Basic properties

Modulus: \(11347\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11347\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(324\)
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 11347.fl

\(\chi_{11347}(327,\cdot)\) \(\chi_{11347}(360,\cdot)\) \(\chi_{11347}(563,\cdot)\) \(\chi_{11347}(668,\cdot)\) \(\chi_{11347}(682,\cdot)\) \(\chi_{11347}(733,\cdot)\) \(\chi_{11347}(964,\cdot)\) \(\chi_{11347}(976,\cdot)\) \(\chi_{11347}(985,\cdot)\) \(\chi_{11347}(1062,\cdot)\) \(\chi_{11347}(1592,\cdot)\) \(\chi_{11347}(1643,\cdot)\) \(\chi_{11347}(1760,\cdot)\) \(\chi_{11347}(2005,\cdot)\) \(\chi_{11347}(2217,\cdot)\) \(\chi_{11347}(2250,\cdot)\) \(\chi_{11347}(2292,\cdot)\) \(\chi_{11347}(2294,\cdot)\) \(\chi_{11347}(2308,\cdot)\) \(\chi_{11347}(2327,\cdot)\) \(\chi_{11347}(2427,\cdot)\) \(\chi_{11347}(2551,\cdot)\) \(\chi_{11347}(2740,\cdot)\) \(\chi_{11347}(2831,\cdot)\) \(\chi_{11347}(2959,\cdot)\) \(\chi_{11347}(2978,\cdot)\) \(\chi_{11347}(3078,\cdot)\) \(\chi_{11347}(3090,\cdot)\) \(\chi_{11347}(3188,\cdot)\) \(\chi_{11347}(3274,\cdot)\) ...

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: $\Q(\zeta_{324})$
Fixed field: Number field defined by a degree 324 polynomial (not computed)

Values on generators

\((6485,8107)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{145}{324}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 11347 }(1062, a) \) \(1\)\(1\)\(e\left(\frac{37}{324}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{37}{162}\right)\)\(e\left(\frac{107}{162}\right)\)\(e\left(\frac{235}{324}\right)\)\(e\left(\frac{37}{108}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{251}{324}\right)\)\(e\left(\frac{13}{81}\right)\)\(e\left(\frac{68}{81}\right)\)
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)
 
\( \chi_{ 11347 }(1062,a) \;\) at \(\;a = \) e.g. 2