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Properties

Label 6496.1899
Modulus $6496$
Conductor $6496$
Order $168$
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(6496, base_ring=CyclotomicField(168)) M = H._module chi = DirichletCharacter(H, M([84,105,56,78]))
 
Copy content gp:[g,chi] = znchar(Mod(1899, 6496))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6496.1899");
 

Basic properties

Modulus: \(6496\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6496\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(168\)
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 6496.ih

\(\chi_{6496}(11,\cdot)\) \(\chi_{6496}(163,\cdot)\) \(\chi_{6496}(235,\cdot)\) \(\chi_{6496}(387,\cdot)\) \(\chi_{6496}(403,\cdot)\) \(\chi_{6496}(627,\cdot)\) \(\chi_{6496}(723,\cdot)\) \(\chi_{6496}(851,\cdot)\) \(\chi_{6496}(891,\cdot)\) \(\chi_{6496}(1059,\cdot)\) \(\chi_{6496}(1187,\cdot)\) \(\chi_{6496}(1355,\cdot)\) \(\chi_{6496}(1523,\cdot)\) \(\chi_{6496}(1899,\cdot)\) \(\chi_{6496}(2067,\cdot)\) \(\chi_{6496}(2235,\cdot)\) \(\chi_{6496}(2363,\cdot)\) \(\chi_{6496}(2531,\cdot)\) \(\chi_{6496}(2571,\cdot)\) \(\chi_{6496}(2699,\cdot)\) \(\chi_{6496}(2795,\cdot)\) \(\chi_{6496}(3019,\cdot)\) \(\chi_{6496}(3035,\cdot)\) \(\chi_{6496}(3187,\cdot)\) \(\chi_{6496}(3259,\cdot)\) \(\chi_{6496}(3411,\cdot)\) \(\chi_{6496}(3483,\cdot)\) \(\chi_{6496}(3635,\cdot)\) \(\chi_{6496}(3651,\cdot)\) \(\chi_{6496}(3875,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((5279,2437,3713,4033)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{3}\right),e\left(\frac{13}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 6496 }(1899, a) \) \(1\)\(1\)\(e\left(\frac{5}{168}\right)\)\(e\left(\frac{85}{168}\right)\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{95}{168}\right)\)\(e\left(\frac{41}{56}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{121}{168}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{1}{84}\right)\)
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)
 
\( \chi_{ 6496 }(1899,a) \;\) at \(\;a = \) e.g. 2