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Properties

Label 4711.475
Modulus $4711$
Conductor $4711$
Order $672$
Real no
Primitive yes
Minimal yes
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(4711, base_ring=CyclotomicField(672)) M = H._module chi = DirichletCharacter(H, M([336,397]))
 
Copy content gp:[g,chi] = znchar(Mod(475, 4711))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4711.475");
 

Basic properties

Modulus: \(4711\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4711\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(672\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 4711.ep

\(\chi_{4711}(20,\cdot)\) \(\chi_{4711}(34,\cdot)\) \(\chi_{4711}(90,\cdot)\) \(\chi_{4711}(132,\cdot)\) \(\chi_{4711}(160,\cdot)\) \(\chi_{4711}(188,\cdot)\) \(\chi_{4711}(237,\cdot)\) \(\chi_{4711}(244,\cdot)\) \(\chi_{4711}(251,\cdot)\) \(\chi_{4711}(258,\cdot)\) \(\chi_{4711}(265,\cdot)\) \(\chi_{4711}(272,\cdot)\) \(\chi_{4711}(279,\cdot)\) \(\chi_{4711}(286,\cdot)\) \(\chi_{4711}(321,\cdot)\) \(\chi_{4711}(342,\cdot)\) \(\chi_{4711}(391,\cdot)\) \(\chi_{4711}(426,\cdot)\) \(\chi_{4711}(433,\cdot)\) \(\chi_{4711}(447,\cdot)\) \(\chi_{4711}(475,\cdot)\) \(\chi_{4711}(531,\cdot)\) \(\chi_{4711}(538,\cdot)\) \(\chi_{4711}(622,\cdot)\) \(\chi_{4711}(629,\cdot)\) \(\chi_{4711}(643,\cdot)\) \(\chi_{4711}(678,\cdot)\) \(\chi_{4711}(692,\cdot)\) \(\chi_{4711}(713,\cdot)\) \(\chi_{4711}(720,\cdot)\) ...

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: $\Q(\zeta_{672})$
Fixed field: Number field defined by a degree 672 polynomial (not computed)

Values on generators

\((1347,2024)\) → \((-1,e\left(\frac{397}{672}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 4711 }(475, a) \) \(1\)\(1\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{47}{168}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{61}{672}\right)\)\(e\left(\frac{269}{336}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{137}{224}\right)\)\(e\left(\frac{643}{672}\right)\)\(e\left(\frac{9}{28}\right)\)
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)
 
\( \chi_{ 4711 }(475,a) \;\) at \(\;a = \) e.g. 2