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Properties

Label 12138.247
Modulus $12138$
Conductor $2023$
Order $408$
Real no
Primitive no
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(12138, base_ring=CyclotomicField(408)) M = H._module chi = DirichletCharacter(H, M([0,136,315]))
 
Copy content gp:[g,chi] = znchar(Mod(247, 12138))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12138.247");
 

Basic properties

Modulus: \(12138\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2023\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(408\)
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: no, induced from \(\chi_{2023}(247,\cdot)\)
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 12138.cw

\(\chi_{12138}(25,\cdot)\) \(\chi_{12138}(121,\cdot)\) \(\chi_{12138}(151,\cdot)\) \(\chi_{12138}(247,\cdot)\) \(\chi_{12138}(331,\cdot)\) \(\chi_{12138}(457,\cdot)\) \(\chi_{12138}(529,\cdot)\) \(\chi_{12138}(655,\cdot)\) \(\chi_{12138}(739,\cdot)\) \(\chi_{12138}(835,\cdot)\) \(\chi_{12138}(865,\cdot)\) \(\chi_{12138}(961,\cdot)\) \(\chi_{12138}(1045,\cdot)\) \(\chi_{12138}(1171,\cdot)\) \(\chi_{12138}(1243,\cdot)\) \(\chi_{12138}(1369,\cdot)\) \(\chi_{12138}(1453,\cdot)\) \(\chi_{12138}(1549,\cdot)\) \(\chi_{12138}(1675,\cdot)\) \(\chi_{12138}(1759,\cdot)\) \(\chi_{12138}(1885,\cdot)\) \(\chi_{12138}(1957,\cdot)\) \(\chi_{12138}(2083,\cdot)\) \(\chi_{12138}(2167,\cdot)\) \(\chi_{12138}(2263,\cdot)\) \(\chi_{12138}(2293,\cdot)\) \(\chi_{12138}(2389,\cdot)\) \(\chi_{12138}(2473,\cdot)\) \(\chi_{12138}(2599,\cdot)\) \(\chi_{12138}(2671,\cdot)\) ...

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: $\Q(\zeta_{408})$
Fixed field: Number field defined by a degree 408 polynomial (not computed)

Values on generators

\((8093,10405,9829)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{105}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 12138 }(247, a) \) \(1\)\(1\)\(e\left(\frac{191}{408}\right)\)\(e\left(\frac{37}{408}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{97}{204}\right)\)\(e\left(\frac{341}{408}\right)\)\(e\left(\frac{191}{204}\right)\)\(e\left(\frac{69}{136}\right)\)\(e\left(\frac{115}{408}\right)\)\(e\left(\frac{107}{408}\right)\)\(e\left(\frac{3}{136}\right)\)
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)
 
\( \chi_{ 12138 }(247,a) \;\) at \(\;a = \) e.g. 2