Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15463, base_ring=CyclotomicField(2162)) M = H._module chi = DirichletCharacter(H, M([0,1921]))
 
Copy content gp:[g,chi] = znchar(Mod(29, 15463))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15463.29");
 

Basic properties

Modulus: \(15463\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2209\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2162\)
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 \(\chi_{2209}(29,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 15463.bb

\(\chi_{15463}(15,\cdot)\) \(\chi_{15463}(22,\cdot)\) \(\chi_{15463}(29,\cdot)\) \(\chi_{15463}(43,\cdot)\) \(\chi_{15463}(57,\cdot)\) \(\chi_{15463}(78,\cdot)\) \(\chi_{15463}(85,\cdot)\) \(\chi_{15463}(92,\cdot)\) \(\chi_{15463}(99,\cdot)\) \(\chi_{15463}(113,\cdot)\) \(\chi_{15463}(120,\cdot)\) \(\chi_{15463}(127,\cdot)\) \(\chi_{15463}(134,\cdot)\) \(\chi_{15463}(176,\cdot)\) \(\chi_{15463}(211,\cdot)\) \(\chi_{15463}(218,\cdot)\) \(\chi_{15463}(232,\cdot)\) \(\chi_{15463}(246,\cdot)\) \(\chi_{15463}(274,\cdot)\) \(\chi_{15463}(302,\cdot)\) \(\chi_{15463}(323,\cdot)\) \(\chi_{15463}(344,\cdot)\) \(\chi_{15463}(351,\cdot)\) \(\chi_{15463}(358,\cdot)\) \(\chi_{15463}(372,\cdot)\) \(\chi_{15463}(386,\cdot)\) \(\chi_{15463}(407,\cdot)\) \(\chi_{15463}(414,\cdot)\) \(\chi_{15463}(421,\cdot)\) \(\chi_{15463}(428,\cdot)\) ...

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: $\Q(\zeta_{1081})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 2162 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((8837,13259)\) → \((1,e\left(\frac{1921}{2162}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 15463 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{85}{1081}\right)\)\(e\left(\frac{879}{1081}\right)\)\(e\left(\frac{170}{1081}\right)\)\(e\left(\frac{1921}{2162}\right)\)\(e\left(\frac{964}{1081}\right)\)\(e\left(\frac{255}{1081}\right)\)\(e\left(\frac{677}{1081}\right)\)\(e\left(\frac{2091}{2162}\right)\)\(e\left(\frac{245}{2162}\right)\)\(e\left(\frac{1049}{1081}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 15463 }(29,a) \;\) at \(\;a = \) e.g. 2