Properties

Label 2209.892
Modulus $2209$
Conductor $2209$
Order $94$
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(94)) M = H._module chi = DirichletCharacter(H, M([85]))
 
Copy content gp:[g,chi] = znchar(Mod(892, 2209))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2209.892");
 

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: \(94\)
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.f

\(\chi_{2209}(46,\cdot)\) \(\chi_{2209}(93,\cdot)\) \(\chi_{2209}(140,\cdot)\) \(\chi_{2209}(187,\cdot)\) \(\chi_{2209}(234,\cdot)\) \(\chi_{2209}(281,\cdot)\) \(\chi_{2209}(328,\cdot)\) \(\chi_{2209}(375,\cdot)\) \(\chi_{2209}(422,\cdot)\) \(\chi_{2209}(469,\cdot)\) \(\chi_{2209}(516,\cdot)\) \(\chi_{2209}(563,\cdot)\) \(\chi_{2209}(610,\cdot)\) \(\chi_{2209}(657,\cdot)\) \(\chi_{2209}(704,\cdot)\) \(\chi_{2209}(751,\cdot)\) \(\chi_{2209}(798,\cdot)\) \(\chi_{2209}(845,\cdot)\) \(\chi_{2209}(892,\cdot)\) \(\chi_{2209}(939,\cdot)\) \(\chi_{2209}(986,\cdot)\) \(\chi_{2209}(1033,\cdot)\) \(\chi_{2209}(1080,\cdot)\) \(\chi_{2209}(1127,\cdot)\) \(\chi_{2209}(1174,\cdot)\) \(\chi_{2209}(1221,\cdot)\) \(\chi_{2209}(1268,\cdot)\) \(\chi_{2209}(1315,\cdot)\) \(\chi_{2209}(1362,\cdot)\) \(\chi_{2209}(1409,\cdot)\) ...

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: $\Q(\zeta_{47})$
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 94 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(5\) → \(e\left(\frac{85}{94}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2209 }(892, a) \) \(-1\)\(1\)\(e\left(\frac{10}{47}\right)\)\(e\left(\frac{26}{47}\right)\)\(e\left(\frac{20}{47}\right)\)\(e\left(\frac{85}{94}\right)\)\(e\left(\frac{36}{47}\right)\)\(e\left(\frac{45}{47}\right)\)\(e\left(\frac{30}{47}\right)\)\(e\left(\frac{5}{47}\right)\)\(e\left(\frac{11}{94}\right)\)\(e\left(\frac{15}{94}\right)\)
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)
 
\( \chi_{ 2209 }(892,a) \;\) at \(\;a = \) e.g. 2