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Properties

Label 1919.245
Modulus $1919$
Conductor $1919$
Order $450$
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(1919, base_ring=CyclotomicField(450)) M = H._module chi = DirichletCharacter(H, M([250,189]))
 
Copy content gp:[g,chi] = znchar(Mod(245, 1919))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1919.245");
 

Basic properties

Modulus: \(1919\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1919\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(450\)
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 1919.bz

\(\chi_{1919}(4,\cdot)\) \(\chi_{1919}(9,\cdot)\) \(\chi_{1919}(23,\cdot)\) \(\chi_{1919}(43,\cdot)\) \(\chi_{1919}(47,\cdot)\) \(\chi_{1919}(82,\cdot)\) \(\chi_{1919}(85,\cdot)\) \(\chi_{1919}(123,\cdot)\) \(\chi_{1919}(131,\cdot)\) \(\chi_{1919}(150,\cdot)\) \(\chi_{1919}(177,\cdot)\) \(\chi_{1919}(206,\cdot)\) \(\chi_{1919}(215,\cdot)\) \(\chi_{1919}(225,\cdot)\) \(\chi_{1919}(232,\cdot)\) \(\chi_{1919}(245,\cdot)\) \(\chi_{1919}(251,\cdot)\) \(\chi_{1919}(272,\cdot)\) \(\chi_{1919}(346,\cdot)\) \(\chi_{1919}(348,\cdot)\) \(\chi_{1919}(367,\cdot)\) \(\chi_{1919}(385,\cdot)\) \(\chi_{1919}(408,\cdot)\) \(\chi_{1919}(424,\cdot)\) \(\chi_{1919}(427,\cdot)\) \(\chi_{1919}(434,\cdot)\) \(\chi_{1919}(453,\cdot)\) \(\chi_{1919}(480,\cdot)\) \(\chi_{1919}(481,\cdot)\) \(\chi_{1919}(500,\cdot)\) ...

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: $\Q(\zeta_{225})$
Fixed field: Number field defined by a degree 450 polynomial (not computed)

Values on generators

\((1617,305)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{21}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1919 }(245, a) \) \(1\)\(1\)\(e\left(\frac{439}{450}\right)\)\(e\left(\frac{91}{450}\right)\)\(e\left(\frac{214}{225}\right)\)\(e\left(\frac{218}{225}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{17}{150}\right)\)\(e\left(\frac{139}{150}\right)\)\(e\left(\frac{91}{225}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{19}{150}\right)\)
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)
 
\( \chi_{ 1919 }(245,a) \;\) at \(\;a = \) e.g. 2