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Properties

Label 4760.1983
Modulus $4760$
Conductor $2380$
Order $48$
Real no
Primitive no
Minimal no
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(4760, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,0,36,16,21]))
 
Copy content gp:[g,chi] = znchar(Mod(1983, 4760))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4760.1983");
 

Basic properties

Modulus: \(4760\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2380\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(48\)
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: no, induced from \(\chi_{2380}(1983,\cdot)\)
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: no
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 4760.kv

\(\chi_{4760}(23,\cdot)\) \(\chi_{4760}(207,\cdot)\) \(\chi_{4760}(823,\cdot)\) \(\chi_{4760}(1383,\cdot)\) \(\chi_{4760}(1983,\cdot)\) \(\chi_{4760}(2207,\cdot)\) \(\chi_{4760}(2487,\cdot)\) \(\chi_{4760}(2543,\cdot)\) \(\chi_{4760}(3327,\cdot)\) \(\chi_{4760}(3343,\cdot)\) \(\chi_{4760}(3567,\cdot)\) \(\chi_{4760}(3607,\cdot)\) \(\chi_{4760}(3847,\cdot)\) \(\chi_{4760}(3903,\cdot)\) \(\chi_{4760}(4223,\cdot)\) \(\chi_{4760}(4687,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((1191,2381,2857,1361,3641)\) → \((-1,1,-i,e\left(\frac{1}{3}\right),e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(19\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 4760 }(1983, a) \) \(-1\)\(1\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{43}{48}\right)\)\(1\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{5}{12}\right)\)
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)
 
\( \chi_{ 4760 }(1983,a) \;\) at \(\;a = \) e.g. 2