↑

Properties

Label 12992.5241
Modulus $12992$
Conductor $6496$
Order $168$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12992, base_ring=CyclotomicField(168)) M = H._module chi = DirichletCharacter(H, M([0,105,140,102]))
 
Copy content gp:[g,chi] = znchar(Mod(5241, 12992))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12992.5241");
 

Basic properties

Modulus: \(12992\)
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: no, induced from \(\chi_{6496}(2805,\cdot)\)
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: no
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 12992.ke

\(\chi_{12992}(73,\cdot)\) \(\chi_{12992}(89,\cdot)\) \(\chi_{12992}(185,\cdot)\) \(\chi_{12992}(201,\cdot)\) \(\chi_{12992}(409,\cdot)\) \(\chi_{12992}(1529,\cdot)\) \(\chi_{12992}(1545,\cdot)\) \(\chi_{12992}(2665,\cdot)\) \(\chi_{12992}(2873,\cdot)\) \(\chi_{12992}(2889,\cdot)\) \(\chi_{12992}(2985,\cdot)\) \(\chi_{12992}(3001,\cdot)\) \(\chi_{12992}(3113,\cdot)\) \(\chi_{12992}(3673,\cdot)\) \(\chi_{12992}(3785,\cdot)\) \(\chi_{12992}(3897,\cdot)\) \(\chi_{12992}(4121,\cdot)\) \(\chi_{12992}(4329,\cdot)\) \(\chi_{12992}(5241,\cdot)\) \(\chi_{12992}(5449,\cdot)\) \(\chi_{12992}(5673,\cdot)\) \(\chi_{12992}(5785,\cdot)\) \(\chi_{12992}(5897,\cdot)\) \(\chi_{12992}(6457,\cdot)\) \(\chi_{12992}(6569,\cdot)\) \(\chi_{12992}(6585,\cdot)\) \(\chi_{12992}(6681,\cdot)\) \(\chi_{12992}(6697,\cdot)\) \(\chi_{12992}(6905,\cdot)\) \(\chi_{12992}(8025,\cdot)\) ...

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

\((11775,2437,3713,4033)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{17}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 12992 }(5241, a) \) \(1\)\(1\)\(e\left(\frac{125}{168}\right)\)\(e\left(\frac{25}{168}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{107}{168}\right)\)\(e\left(\frac{45}{56}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{168}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{25}{84}\right)\)
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)
 
\( \chi_{ 12992 }(5241,a) \;\) at \(\;a = \) e.g. 2