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Properties

Label 2557.1068
Modulus $2557$
Conductor $2557$
Order $2556$
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(2557, base_ring=CyclotomicField(2556)) M = H._module chi = DirichletCharacter(H, M([827]))
 
Copy content gp:[g,chi] = znchar(Mod(1068, 2557))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2557.1068");
 

Basic properties

Modulus: \(2557\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2557\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2556\)
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 2557.r

\(\chi_{2557}(2,\cdot)\) \(\chi_{2557}(5,\cdot)\) \(\chi_{2557}(6,\cdot)\) \(\chi_{2557}(15,\cdot)\) \(\chi_{2557}(17,\cdot)\) \(\chi_{2557}(24,\cdot)\) \(\chi_{2557}(31,\cdot)\) \(\chi_{2557}(32,\cdot)\) \(\chi_{2557}(41,\cdot)\) \(\chi_{2557}(42,\cdot)\) \(\chi_{2557}(43,\cdot)\) \(\chi_{2557}(47,\cdot)\) \(\chi_{2557}(51,\cdot)\) \(\chi_{2557}(54,\cdot)\) \(\chi_{2557}(56,\cdot)\) \(\chi_{2557}(60,\cdot)\) \(\chi_{2557}(66,\cdot)\) \(\chi_{2557}(67,\cdot)\) \(\chi_{2557}(72,\cdot)\) \(\chi_{2557}(78,\cdot)\) \(\chi_{2557}(80,\cdot)\) \(\chi_{2557}(83,\cdot)\) \(\chi_{2557}(88,\cdot)\) \(\chi_{2557}(89,\cdot)\) \(\chi_{2557}(98,\cdot)\) \(\chi_{2557}(103,\cdot)\) \(\chi_{2557}(104,\cdot)\) \(\chi_{2557}(105,\cdot)\) \(\chi_{2557}(106,\cdot)\) \(\chi_{2557}(107,\cdot)\) ...

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: $\Q(\zeta_{2556})$
Fixed field: Number field defined by a degree 2556 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{827}{2556}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2557 }(1068, a) \) \(-1\)\(1\)\(e\left(\frac{827}{2556}\right)\)\(e\left(\frac{35}{639}\right)\)\(e\left(\frac{827}{1278}\right)\)\(e\left(\frac{365}{2556}\right)\)\(e\left(\frac{967}{2556}\right)\)\(e\left(\frac{887}{1278}\right)\)\(e\left(\frac{827}{852}\right)\)\(e\left(\frac{70}{639}\right)\)\(e\left(\frac{298}{639}\right)\)\(e\left(\frac{334}{639}\right)\)
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)
 
\( \chi_{ 2557 }(1068,a) \;\) at \(\;a = \) e.g. 2