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Properties

Label 11323.578
Modulus $11323$
Conductor $11323$
Order $1716$
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(11323, base_ring=CyclotomicField(1716)) M = H._module chi = DirichletCharacter(H, M([1507,1638]))
 
Copy content gp:[g,chi] = znchar(Mod(578, 11323))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11323.578");
 

Basic properties

Modulus: \(11323\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11323\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1716\)
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 11323.eo

\(\chi_{11323}(45,\cdot)\) \(\chi_{11323}(58,\cdot)\) \(\chi_{11323}(72,\cdot)\) \(\chi_{11323}(110,\cdot)\) \(\chi_{11323}(119,\cdot)\) \(\chi_{11323}(137,\cdot)\) \(\chi_{11323}(176,\cdot)\) \(\chi_{11323}(206,\cdot)\) \(\chi_{11323}(228,\cdot)\) \(\chi_{11323}(253,\cdot)\) \(\chi_{11323}(254,\cdot)\) \(\chi_{11323}(271,\cdot)\) \(\chi_{11323}(310,\cdot)\) \(\chi_{11323}(340,\cdot)\) \(\chi_{11323}(362,\cdot)\) \(\chi_{11323}(388,\cdot)\) \(\chi_{11323}(405,\cdot)\) \(\chi_{11323}(410,\cdot)\) \(\chi_{11323}(444,\cdot)\) \(\chi_{11323}(474,\cdot)\) \(\chi_{11323}(496,\cdot)\) \(\chi_{11323}(514,\cdot)\) \(\chi_{11323}(522,\cdot)\) \(\chi_{11323}(527,\cdot)\) \(\chi_{11323}(539,\cdot)\) \(\chi_{11323}(544,\cdot)\) \(\chi_{11323}(578,\cdot)\) \(\chi_{11323}(579,\cdot)\) \(\chi_{11323}(630,\cdot)\) \(\chi_{11323}(648,\cdot)\) ...

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: $\Q(\zeta_{1716})$
Fixed field: Number field defined by a degree 1716 polynomial (not computed)

Values on generators

\((7438,3888)\) → \((e\left(\frac{137}{156}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11323 }(578, a) \) \(1\)\(1\)\(e\left(\frac{1429}{1716}\right)\)\(e\left(\frac{107}{858}\right)\)\(e\left(\frac{571}{858}\right)\)\(e\left(\frac{127}{572}\right)\)\(e\left(\frac{1643}{1716}\right)\)\(e\left(\frac{1583}{1716}\right)\)\(e\left(\frac{285}{572}\right)\)\(e\left(\frac{107}{429}\right)\)\(e\left(\frac{47}{858}\right)\)\(e\left(\frac{1327}{1716}\right)\)
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)
 
\( \chi_{ 11323 }(578,a) \;\) at \(\;a = \) e.g. 2