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Properties

Label 7057.26
Modulus $7057$
Conductor $7057$
Order $784$
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(7057, base_ring=CyclotomicField(784)) M = H._module chi = DirichletCharacter(H, M([229]))
 
Copy content gp:[g,chi] = znchar(Mod(26, 7057))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7057.26");
 

Basic properties

Modulus: \(7057\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7057\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(784\)
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 7057.bl

\(\chi_{7057}(13,\cdot)\) \(\chi_{7057}(26,\cdot)\) \(\chi_{7057}(52,\cdot)\) \(\chi_{7057}(77,\cdot)\) \(\chi_{7057}(83,\cdot)\) \(\chi_{7057}(85,\cdot)\) \(\chi_{7057}(87,\cdot)\) \(\chi_{7057}(104,\cdot)\) \(\chi_{7057}(106,\cdot)\) \(\chi_{7057}(133,\cdot)\) \(\chi_{7057}(154,\cdot)\) \(\chi_{7057}(166,\cdot)\) \(\chi_{7057}(170,\cdot)\) \(\chi_{7057}(174,\cdot)\) \(\chi_{7057}(208,\cdot)\) \(\chi_{7057}(212,\cdot)\) \(\chi_{7057}(251,\cdot)\) \(\chi_{7057}(266,\cdot)\) \(\chi_{7057}(275,\cdot)\) \(\chi_{7057}(331,\cdot)\) \(\chi_{7057}(332,\cdot)\) \(\chi_{7057}(340,\cdot)\) \(\chi_{7057}(348,\cdot)\) \(\chi_{7057}(401,\cdot)\) \(\chi_{7057}(424,\cdot)\) \(\chi_{7057}(475,\cdot)\) \(\chi_{7057}(502,\cdot)\) \(\chi_{7057}(532,\cdot)\) \(\chi_{7057}(533,\cdot)\) \(\chi_{7057}(543,\cdot)\) ...

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: $\Q(\zeta_{784})$
Fixed field: Number field defined by a degree 784 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{229}{784}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 7057 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{139}{392}\right)\)\(e\left(\frac{87}{98}\right)\)\(e\left(\frac{139}{196}\right)\)\(e\left(\frac{229}{784}\right)\)\(e\left(\frac{95}{392}\right)\)\(e\left(\frac{15}{98}\right)\)\(e\left(\frac{25}{392}\right)\)\(e\left(\frac{38}{49}\right)\)\(e\left(\frac{507}{784}\right)\)\(e\left(\frac{381}{784}\right)\)
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)
 
\( \chi_{ 7057 }(26,a) \;\) at \(\;a = \) e.g. 2