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Properties

Label 6169.286
Modulus $6169$
Conductor $6169$
Order $990$
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(6169, base_ring=CyclotomicField(990)) M = H._module chi = DirichletCharacter(H, M([924,355]))
 
Copy content gp:[g,chi] = znchar(Mod(286, 6169))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6169.286");
 

Basic properties

Modulus: \(6169\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6169\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(990\)
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 6169.ew

\(\chi_{6169}(38,\cdot)\) \(\chi_{6169}(69,\cdot)\) \(\chi_{6169}(71,\cdot)\) \(\chi_{6169}(134,\cdot)\) \(\chi_{6169}(164,\cdot)\) \(\chi_{6169}(205,\cdot)\) \(\chi_{6169}(267,\cdot)\) \(\chi_{6169}(286,\cdot)\) \(\chi_{6169}(298,\cdot)\) \(\chi_{6169}(307,\cdot)\) \(\chi_{6169}(319,\cdot)\) \(\chi_{6169}(351,\cdot)\) \(\chi_{6169}(382,\cdot)\) \(\chi_{6169}(413,\cdot)\) \(\chi_{6169}(452,\cdot)\) \(\chi_{6169}(475,\cdot)\) \(\chi_{6169}(503,\cdot)\) \(\chi_{6169}(516,\cdot)\) \(\chi_{6169}(547,\cdot)\) \(\chi_{6169}(565,\cdot)\) \(\chi_{6169}(568,\cdot)\) \(\chi_{6169}(577,\cdot)\) \(\chi_{6169}(603,\cdot)\) \(\chi_{6169}(627,\cdot)\) \(\chi_{6169}(665,\cdot)\) \(\chi_{6169}(692,\cdot)\) \(\chi_{6169}(696,\cdot)\) \(\chi_{6169}(710,\cdot)\) \(\chi_{6169}(731,\cdot)\) \(\chi_{6169}(751,\cdot)\) ...

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: $\Q(\zeta_{495})$
Fixed field: Number field defined by a degree 990 polynomial (not computed)

Values on generators

\((4777,1396)\) → \((e\left(\frac{14}{15}\right),e\left(\frac{71}{198}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6169 }(286, a) \) \(-1\)\(1\)\(e\left(\frac{203}{495}\right)\)\(e\left(\frac{289}{990}\right)\)\(e\left(\frac{406}{495}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{139}{198}\right)\)\(e\left(\frac{26}{495}\right)\)\(e\left(\frac{38}{165}\right)\)\(e\left(\frac{289}{495}\right)\)\(e\left(\frac{278}{495}\right)\)\(e\left(\frac{79}{330}\right)\)
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)
 
\( \chi_{ 6169 }(286,a) \;\) at \(\;a = \) e.g. 2