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Properties

Label 2164.1199
Modulus $2164$
Conductor $2164$
Order $540$
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(2164, base_ring=CyclotomicField(540)) M = H._module chi = DirichletCharacter(H, M([270,341]))
 
Copy content gp:[g,chi] = znchar(Mod(1199, 2164))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2164.1199");
 

Basic properties

Modulus: \(2164\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2164\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(540\)
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 2164.bu

\(\chi_{2164}(51,\cdot)\) \(\chi_{2164}(55,\cdot)\) \(\chi_{2164}(59,\cdot)\) \(\chi_{2164}(67,\cdot)\) \(\chi_{2164}(83,\cdot)\) \(\chi_{2164}(87,\cdot)\) \(\chi_{2164}(91,\cdot)\) \(\chi_{2164}(99,\cdot)\) \(\chi_{2164}(107,\cdot)\) \(\chi_{2164}(127,\cdot)\) \(\chi_{2164}(131,\cdot)\) \(\chi_{2164}(163,\cdot)\) \(\chi_{2164}(183,\cdot)\) \(\chi_{2164}(195,\cdot)\) \(\chi_{2164}(199,\cdot)\) \(\chi_{2164}(223,\cdot)\) \(\chi_{2164}(259,\cdot)\) \(\chi_{2164}(263,\cdot)\) \(\chi_{2164}(267,\cdot)\) \(\chi_{2164}(271,\cdot)\) \(\chi_{2164}(283,\cdot)\) \(\chi_{2164}(291,\cdot)\) \(\chi_{2164}(323,\cdot)\) \(\chi_{2164}(331,\cdot)\) \(\chi_{2164}(335,\cdot)\) \(\chi_{2164}(383,\cdot)\) \(\chi_{2164}(391,\cdot)\) \(\chi_{2164}(403,\cdot)\) \(\chi_{2164}(415,\cdot)\) \(\chi_{2164}(427,\cdot)\) ...

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: $\Q(\zeta_{540})$
Fixed field: Number field defined by a degree 540 polynomial (not computed)

Values on generators

\((1083,1625)\) → \((-1,e\left(\frac{341}{540}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2164 }(1199, a) \) \(1\)\(1\)\(e\left(\frac{47}{270}\right)\)\(e\left(\frac{29}{135}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{47}{135}\right)\)\(e\left(\frac{85}{108}\right)\)\(e\left(\frac{533}{540}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{22}{135}\right)\)\(e\left(\frac{161}{270}\right)\)
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)
 
\( \chi_{ 2164 }(1199,a) \;\) at \(\;a = \) e.g. 2