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Properties

Label 115089.245
Modulus $115089$
Conductor $115089$
Order $17628$
Real no
Primitive yes
Minimal yes
Parity odd

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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(115089, base_ring=CyclotomicField(17628)) M = H._module chi = DirichletCharacter(H, M([8814,7571,7254]))
 
Copy content gp:[g,chi] = znchar(Mod(245, 115089))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("115089.245");
 

Basic properties

Modulus: \(115089\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(115089\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(17628\)
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 115089.dq

\(\chi_{115089}(2,\cdot)\) \(\chi_{115089}(20,\cdot)\) \(\chi_{115089}(32,\cdot)\) \(\chi_{115089}(41,\cdot)\) \(\chi_{115089}(50,\cdot)\) \(\chi_{115089}(98,\cdot)\) \(\chi_{115089}(119,\cdot)\) \(\chi_{115089}(128,\cdot)\) \(\chi_{115089}(137,\cdot)\) \(\chi_{115089}(149,\cdot)\) \(\chi_{115089}(158,\cdot)\) \(\chi_{115089}(197,\cdot)\) \(\chi_{115089}(206,\cdot)\) \(\chi_{115089}(215,\cdot)\) \(\chi_{115089}(245,\cdot)\) \(\chi_{115089}(266,\cdot)\) \(\chi_{115089}(293,\cdot)\) \(\chi_{115089}(323,\cdot)\) \(\chi_{115089}(332,\cdot)\) \(\chi_{115089}(344,\cdot)\) \(\chi_{115089}(353,\cdot)\) \(\chi_{115089}(362,\cdot)\) \(\chi_{115089}(383,\cdot)\) \(\chi_{115089}(392,\cdot)\) \(\chi_{115089}(401,\cdot)\) \(\chi_{115089}(410,\cdot)\) \(\chi_{115089}(431,\cdot)\) \(\chi_{115089}(500,\cdot)\) \(\chi_{115089}(509,\cdot)\) \(\chi_{115089}(548,\cdot)\) ...

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: $\Q(\zeta_{17628})$
Fixed field: Number field defined by a degree 17628 polynomial (not computed)

Values on generators

\((76727,23155,15211)\) → \((-1,e\left(\frac{67}{156}\right),e\left(\frac{93}{226}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 115089 }(245, a) \) \(-1\)\(1\)\(e\left(\frac{6011}{17628}\right)\)\(e\left(\frac{6011}{8814}\right)\)\(e\left(\frac{5241}{5876}\right)\)\(e\left(\frac{5761}{17628}\right)\)\(e\left(\frac{135}{5876}\right)\)\(e\left(\frac{2053}{8814}\right)\)\(e\left(\frac{4571}{17628}\right)\)\(e\left(\frac{981}{1469}\right)\)\(e\left(\frac{1604}{4407}\right)\)\(e\left(\frac{8321}{8814}\right)\)
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)
 
\( \chi_{ 115089 }(245,a) \;\) at \(\;a = \) e.g. 2