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Properties

Label 1169.1027
Modulus $1169$
Conductor $1169$
Order $498$
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(1169, base_ring=CyclotomicField(498)) M = H._module chi = DirichletCharacter(H, M([415,6]))
 
Copy content gp:[g,chi] = znchar(Mod(1027, 1169))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1169.1027");
 

Basic properties

Modulus: \(1169\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1169\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(498\)
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 1169.n

\(\chi_{1169}(3,\cdot)\) \(\chi_{1169}(12,\cdot)\) \(\chi_{1169}(19,\cdot)\) \(\chi_{1169}(24,\cdot)\) \(\chi_{1169}(31,\cdot)\) \(\chi_{1169}(33,\cdot)\) \(\chi_{1169}(38,\cdot)\) \(\chi_{1169}(47,\cdot)\) \(\chi_{1169}(54,\cdot)\) \(\chi_{1169}(61,\cdot)\) \(\chi_{1169}(66,\cdot)\) \(\chi_{1169}(75,\cdot)\) \(\chi_{1169}(87,\cdot)\) \(\chi_{1169}(89,\cdot)\) \(\chi_{1169}(94,\cdot)\) \(\chi_{1169}(96,\cdot)\) \(\chi_{1169}(108,\cdot)\) \(\chi_{1169}(115,\cdot)\) \(\chi_{1169}(122,\cdot)\) \(\chi_{1169}(124,\cdot)\) \(\chi_{1169}(150,\cdot)\) \(\chi_{1169}(152,\cdot)\) \(\chi_{1169}(157,\cdot)\) \(\chi_{1169}(171,\cdot)\) \(\chi_{1169}(173,\cdot)\) \(\chi_{1169}(178,\cdot)\) \(\chi_{1169}(185,\cdot)\) \(\chi_{1169}(192,\cdot)\) \(\chi_{1169}(194,\cdot)\) \(\chi_{1169}(199,\cdot)\) ...

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: $\Q(\zeta_{249})$
Fixed field: Number field defined by a degree 498 polynomial (not computed)

Values on generators

\((836,673)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1169 }(1027, a) \) \(-1\)\(1\)\(e\left(\frac{37}{249}\right)\)\(e\left(\frac{481}{498}\right)\)\(e\left(\frac{74}{249}\right)\)\(e\left(\frac{89}{498}\right)\)\(e\left(\frac{19}{166}\right)\)\(e\left(\frac{37}{83}\right)\)\(e\left(\frac{232}{249}\right)\)\(e\left(\frac{163}{498}\right)\)\(e\left(\frac{167}{249}\right)\)\(e\left(\frac{131}{498}\right)\)
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)
 
\( \chi_{ 1169 }(1027,a) \;\) at \(\;a = \) e.g. 2