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Properties

Label 1509.95
Modulus $1509$
Conductor $1509$
Order $502$
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(1509, base_ring=CyclotomicField(502)) M = H._module chi = DirichletCharacter(H, M([251,238]))
 
Copy content gp:[g,chi] = znchar(Mod(95, 1509))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1509.95");
 

Basic properties

Modulus: \(1509\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1509\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(502\)
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 1509.h

\(\chi_{1509}(2,\cdot)\) \(\chi_{1509}(8,\cdot)\) \(\chi_{1509}(11,\cdot)\) \(\chi_{1509}(14,\cdot)\) \(\chi_{1509}(23,\cdot)\) \(\chi_{1509}(26,\cdot)\) \(\chi_{1509}(32,\cdot)\) \(\chi_{1509}(44,\cdot)\) \(\chi_{1509}(47,\cdot)\) \(\chi_{1509}(50,\cdot)\) \(\chi_{1509}(56,\cdot)\) \(\chi_{1509}(59,\cdot)\) \(\chi_{1509}(77,\cdot)\) \(\chi_{1509}(83,\cdot)\) \(\chi_{1509}(86,\cdot)\) \(\chi_{1509}(92,\cdot)\) \(\chi_{1509}(95,\cdot)\) \(\chi_{1509}(98,\cdot)\) \(\chi_{1509}(104,\cdot)\) \(\chi_{1509}(113,\cdot)\) \(\chi_{1509}(122,\cdot)\) \(\chi_{1509}(128,\cdot)\) \(\chi_{1509}(131,\cdot)\) \(\chi_{1509}(134,\cdot)\) \(\chi_{1509}(143,\cdot)\) \(\chi_{1509}(146,\cdot)\) \(\chi_{1509}(155,\cdot)\) \(\chi_{1509}(158,\cdot)\) \(\chi_{1509}(161,\cdot)\) \(\chi_{1509}(170,\cdot)\) ...

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: $\Q(\zeta_{251})$
Fixed field: Number field defined by a degree 502 polynomial (not computed)

Values on generators

\((1007,508)\) → \((-1,e\left(\frac{119}{251}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1509 }(95, a) \) \(-1\)\(1\)\(e\left(\frac{135}{502}\right)\)\(e\left(\frac{135}{251}\right)\)\(e\left(\frac{489}{502}\right)\)\(e\left(\frac{194}{251}\right)\)\(e\left(\frac{405}{502}\right)\)\(e\left(\frac{61}{251}\right)\)\(e\left(\frac{207}{502}\right)\)\(e\left(\frac{198}{251}\right)\)\(e\left(\frac{21}{502}\right)\)\(e\left(\frac{19}{251}\right)\)
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)
 
\( \chi_{ 1509 }(95,a) \;\) at \(\;a = \) e.g. 2