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Properties

Label 152269.51695
Modulus $152269$
Conductor $11713$
Order $312$
Real no
Primitive no
Minimal no
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(152269, base_ring=CyclotomicField(312)) M = H._module chi = DirichletCharacter(H, M([286,117,294]))
 
Copy content gp:[g,chi] = znchar(Mod(51695, 152269))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152269.51695");
 

Basic properties

Modulus: \(152269\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11713\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(312\)
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: no, induced from \(\chi_{11713}(4843,\cdot)\)
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: no
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 152269.cls

\(\chi_{152269}(427,\cdot)\) \(\chi_{152269}(756,\cdot)\) \(\chi_{152269}(2854,\cdot)\) \(\chi_{152269}(4751,\cdot)\) \(\chi_{152269}(5727,\cdot)\) \(\chi_{152269}(12248,\cdot)\) \(\chi_{152269}(13370,\cdot)\) \(\chi_{152269}(15468,\cdot)\) \(\chi_{152269}(17994,\cdot)\) \(\chi_{152269}(18341,\cdot)\) \(\chi_{152269}(24862,\cdot)\) \(\chi_{152269}(26444,\cdot)\) \(\chi_{152269}(29317,\cdot)\) \(\chi_{152269}(29486,\cdot)\) \(\chi_{152269}(30608,\cdot)\) \(\chi_{152269}(33781,\cdot)\) \(\chi_{152269}(36654,\cdot)\) \(\chi_{152269}(38105,\cdot)\) \(\chi_{152269}(39058,\cdot)\) \(\chi_{152269}(41931,\cdot)\) \(\chi_{152269}(42100,\cdot)\) \(\chi_{152269}(46395,\cdot)\) \(\chi_{152269}(46555,\cdot)\) \(\chi_{152269}(49268,\cdot)\) \(\chi_{152269}(50719,\cdot)\) \(\chi_{152269}(51695,\cdot)\) \(\chi_{152269}(54568,\cdot)\) \(\chi_{152269}(56765,\cdot)\) \(\chi_{152269}(58047,\cdot)\) \(\chi_{152269}(58216,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((149567,143313,83318)\) → \((e\left(\frac{11}{12}\right),e\left(\frac{3}{8}\right),e\left(\frac{49}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 152269 }(51695, a) \) \(1\)\(1\)\(e\left(\frac{17}{156}\right)\)\(e\left(\frac{19}{312}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{43}{104}\right)\)\(e\left(\frac{53}{312}\right)\)\(e\left(\frac{125}{312}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{19}{156}\right)\)\(e\left(\frac{163}{312}\right)\)\(e\left(\frac{217}{312}\right)\)
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)
 
\( \chi_{ 152269 }(51695,a) \;\) at \(\;a = \) e.g. 2