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Properties

Label 11347.3771
Modulus $11347$
Conductor $11347$
Order $270$
Real no
Primitive yes
Minimal yes
Parity odd

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(270)) M = H._module chi = DirichletCharacter(H, M([225,89]))
 
Copy content gp:[g,chi] = znchar(Mod(3771, 11347))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11347.3771");
 

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: \(270\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 11347.fg

\(\chi_{11347}(143,\cdot)\) \(\chi_{11347}(192,\cdot)\) \(\chi_{11347}(208,\cdot)\) \(\chi_{11347}(402,\cdot)\) \(\chi_{11347}(465,\cdot)\) \(\chi_{11347}(488,\cdot)\) \(\chi_{11347}(619,\cdot)\) \(\chi_{11347}(654,\cdot)\) \(\chi_{11347}(796,\cdot)\) \(\chi_{11347}(1102,\cdot)\) \(\chi_{11347}(1272,\cdot)\) \(\chi_{11347}(1293,\cdot)\) \(\chi_{11347}(1496,\cdot)\) \(\chi_{11347}(1587,\cdot)\) \(\chi_{11347}(1816,\cdot)\) \(\chi_{11347}(1942,\cdot)\) \(\chi_{11347}(2047,\cdot)\) \(\chi_{11347}(2124,\cdot)\) \(\chi_{11347}(2245,\cdot)\) \(\chi_{11347}(2637,\cdot)\) \(\chi_{11347}(2908,\cdot)\) \(\chi_{11347}(3085,\cdot)\) \(\chi_{11347}(3384,\cdot)\) \(\chi_{11347}(3470,\cdot)\) \(\chi_{11347}(3666,\cdot)\) \(\chi_{11347}(3671,\cdot)\) \(\chi_{11347}(3673,\cdot)\) \(\chi_{11347}(3771,\cdot)\) \(\chi_{11347}(3804,\cdot)\) \(\chi_{11347}(3818,\cdot)\) ...

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: $\Q(\zeta_{135})$
Fixed field: Number field defined by a degree 270 polynomial (not computed)

Values on generators

\((6485,8107)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{89}{270}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 11347 }(3771, a) \) \(-1\)\(1\)\(e\left(\frac{269}{270}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{134}{135}\right)\)\(e\left(\frac{119}{270}\right)\)\(e\left(\frac{13}{135}\right)\)\(e\left(\frac{89}{90}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{59}{135}\right)\)\(e\left(\frac{28}{135}\right)\)\(e\left(\frac{5}{54}\right)\)
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)
 
\( \chi_{ 11347 }(3771,a) \;\) at \(\;a = \) e.g. 2