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Properties

Label 246924.47
Modulus $246924$
Conductor $246924$
Order $6498$
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(246924, base_ring=CyclotomicField(6498)) M = H._module chi = DirichletCharacter(H, M([3249,1083,2420]))
 
Copy content gp:[g,chi] = znchar(Mod(47, 246924))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("246924.47");
 

Basic properties

Modulus: \(246924\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(246924\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(6498\)
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 246924.gz

\(\chi_{246924}(47,\cdot)\) \(\chi_{246924}(131,\cdot)\) \(\chi_{246924}(347,\cdot)\) \(\chi_{246924}(443,\cdot)\) \(\chi_{246924}(479,\cdot)\) \(\chi_{246924}(491,\cdot)\) \(\chi_{246924}(731,\cdot)\) \(\chi_{246924}(815,\cdot)\) \(\chi_{246924}(1031,\cdot)\) \(\chi_{246924}(1127,\cdot)\) \(\chi_{246924}(1163,\cdot)\) \(\chi_{246924}(1175,\cdot)\) \(\chi_{246924}(1415,\cdot)\) \(\chi_{246924}(1499,\cdot)\)

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: $\Q(\zeta_{3249})$
Fixed field: Number field defined by a degree 6498 polynomial (not computed)

Values on generators

\((123463,192053,116605)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1210}{3249}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 246924 }(47, a) \) \(1\)\(1\)\(e\left(\frac{2555}{6498}\right)\)\(e\left(\frac{863}{2166}\right)\)\(e\left(\frac{331}{361}\right)\)\(e\left(\frac{1256}{3249}\right)\)\(e\left(\frac{1943}{6498}\right)\)\(e\left(\frac{2468}{3249}\right)\)\(e\left(\frac{2555}{3249}\right)\)\(e\left(\frac{2209}{6498}\right)\)\(e\left(\frac{71}{722}\right)\)\(e\left(\frac{2572}{3249}\right)\)
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)
 
\( \chi_{ 246924 }(47,a) \;\) at \(\;a = \) e.g. 2