Properties

Label 2209.29
Modulus $2209$
Conductor $2209$
Order $2162$
Real no
Primitive yes
Minimal yes
Parity odd

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(2209, base_ring=CyclotomicField(2162)) M = H._module chi = DirichletCharacter(H, M([1921]))
 
Copy content gp:[g,chi] = znchar(Mod(29, 2209))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2209.29");
 

Basic properties

Modulus: \(2209\)
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: 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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2209.h

\(\chi_{2209}(5,\cdot)\) \(\chi_{2209}(10,\cdot)\) \(\chi_{2209}(11,\cdot)\) \(\chi_{2209}(13,\cdot)\) \(\chi_{2209}(15,\cdot)\) \(\chi_{2209}(19,\cdot)\) \(\chi_{2209}(20,\cdot)\) \(\chi_{2209}(22,\cdot)\) \(\chi_{2209}(23,\cdot)\) \(\chi_{2209}(26,\cdot)\) \(\chi_{2209}(29,\cdot)\) \(\chi_{2209}(30,\cdot)\) \(\chi_{2209}(31,\cdot)\) \(\chi_{2209}(33,\cdot)\) \(\chi_{2209}(35,\cdot)\) \(\chi_{2209}(38,\cdot)\) \(\chi_{2209}(39,\cdot)\) \(\chi_{2209}(40,\cdot)\) \(\chi_{2209}(41,\cdot)\) \(\chi_{2209}(43,\cdot)\) \(\chi_{2209}(44,\cdot)\) \(\chi_{2209}(45,\cdot)\) \(\chi_{2209}(52,\cdot)\) \(\chi_{2209}(57,\cdot)\) \(\chi_{2209}(58,\cdot)\) \(\chi_{2209}(60,\cdot)\) \(\chi_{2209}(62,\cdot)\) \(\chi_{2209}(66,\cdot)\) \(\chi_{2209}(69,\cdot)\) \(\chi_{2209}(70,\cdot)\) ...

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

\(5\) → \(e\left(\frac{1921}{2162}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2209 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{85}{1081}\right)\)\(e\left(\frac{879}{1081}\right)\)\(e\left(\frac{170}{1081}\right)\)\(e\left(\frac{1921}{2162}\right)\)\(e\left(\frac{964}{1081}\right)\)\(e\left(\frac{77}{1081}\right)\)\(e\left(\frac{255}{1081}\right)\)\(e\left(\frac{677}{1081}\right)\)\(e\left(\frac{2091}{2162}\right)\)\(e\left(\frac{245}{2162}\right)\)
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)
 
\( \chi_{ 2209 }(29,a) \;\) at \(\;a = \) e.g. 2