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Properties

Label 2515.393
Modulus $2515$
Conductor $2515$
Order $1004$
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(2515, base_ring=CyclotomicField(1004)) M = H._module chi = DirichletCharacter(H, M([753,992]))
 
Copy content gp:[g,chi] = znchar(Mod(393, 2515))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2515.393");
 

Basic properties

Modulus: \(2515\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2515\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1004\)
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 2515.l

\(\chi_{2515}(2,\cdot)\) \(\chi_{2515}(3,\cdot)\) \(\chi_{2515}(7,\cdot)\) \(\chi_{2515}(8,\cdot)\) \(\chi_{2515}(12,\cdot)\) \(\chi_{2515}(13,\cdot)\) \(\chi_{2515}(18,\cdot)\) \(\chi_{2515}(22,\cdot)\) \(\chi_{2515}(23,\cdot)\) \(\chi_{2515}(27,\cdot)\) \(\chi_{2515}(28,\cdot)\) \(\chi_{2515}(32,\cdot)\) \(\chi_{2515}(33,\cdot)\) \(\chi_{2515}(42,\cdot)\) \(\chi_{2515}(43,\cdot)\) \(\chi_{2515}(47,\cdot)\) \(\chi_{2515}(48,\cdot)\) \(\chi_{2515}(52,\cdot)\) \(\chi_{2515}(63,\cdot)\) \(\chi_{2515}(67,\cdot)\) \(\chi_{2515}(72,\cdot)\) \(\chi_{2515}(73,\cdot)\) \(\chi_{2515}(77,\cdot)\) \(\chi_{2515}(78,\cdot)\) \(\chi_{2515}(83,\cdot)\) \(\chi_{2515}(88,\cdot)\) \(\chi_{2515}(92,\cdot)\) \(\chi_{2515}(97,\cdot)\) \(\chi_{2515}(98,\cdot)\) \(\chi_{2515}(108,\cdot)\) ...

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: $\Q(\zeta_{1004})$
Fixed field: Number field defined by a degree 1004 polynomial (not computed)

Values on generators

\((1007,1011)\) → \((-i,e\left(\frac{248}{251}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2515 }(393, a) \) \(-1\)\(1\)\(e\left(\frac{337}{1004}\right)\)\(e\left(\frac{387}{1004}\right)\)\(e\left(\frac{337}{502}\right)\)\(e\left(\frac{181}{251}\right)\)\(e\left(\frac{725}{1004}\right)\)\(e\left(\frac{7}{1004}\right)\)\(e\left(\frac{387}{502}\right)\)\(e\left(\frac{125}{251}\right)\)\(e\left(\frac{57}{1004}\right)\)\(e\left(\frac{771}{1004}\right)\)
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)
 
\( \chi_{ 2515 }(393,a) \;\) at \(\;a = \) e.g. 2