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Properties

Label 50575.1017
Modulus $50575$
Conductor $50575$
Order $4080$
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(50575, base_ring=CyclotomicField(4080)) M = H._module chi = DirichletCharacter(H, M([2652,1360,1575]))
 
Copy content gp:[g,chi] = znchar(Mod(1017, 50575))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50575.1017");
 

Basic properties

Modulus: \(50575\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(50575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4080\)
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 50575.la

\(\chi_{50575}(23,\cdot)\) \(\chi_{50575}(37,\cdot)\) \(\chi_{50575}(58,\cdot)\) \(\chi_{50575}(163,\cdot)\) \(\chi_{50575}(198,\cdot)\) \(\chi_{50575}(228,\cdot)\) \(\chi_{50575}(277,\cdot)\) \(\chi_{50575}(333,\cdot)\) \(\chi_{50575}(352,\cdot)\) \(\chi_{50575}(422,\cdot)\) \(\chi_{50575}(522,\cdot)\) \(\chi_{50575}(592,\cdot)\) \(\chi_{50575}(702,\cdot)\) \(\chi_{50575}(758,\cdot)\) \(\chi_{50575}(788,\cdot)\) \(\chi_{50575}(823,\cdot)\) \(\chi_{50575}(872,\cdot)\) \(\chi_{50575}(928,\cdot)\) \(\chi_{50575}(947,\cdot)\) \(\chi_{50575}(963,\cdot)\) \(\chi_{50575}(1017,\cdot)\) \(\chi_{50575}(1117,\cdot)\) \(\chi_{50575}(1187,\cdot)\) \(\chi_{50575}(1213,\cdot)\) \(\chi_{50575}(1227,\cdot)\) \(\chi_{50575}(1248,\cdot)\) \(\chi_{50575}(1297,\cdot)\) \(\chi_{50575}(1353,\cdot)\) \(\chi_{50575}(1383,\cdot)\) \(\chi_{50575}(1388,\cdot)\) ...

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: $\Q(\zeta_{4080})$
Fixed field: Number field defined by a degree 4080 polynomial (not computed)

Values on generators

\((24277,14451,38151)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{105}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 50575 }(1017, a) \) \(1\)\(1\)\(e\left(\frac{1351}{2040}\right)\)\(e\left(\frac{1099}{4080}\right)\)\(e\left(\frac{331}{1020}\right)\)\(e\left(\frac{1267}{1360}\right)\)\(e\left(\frac{671}{680}\right)\)\(e\left(\frac{1099}{2040}\right)\)\(e\left(\frac{2497}{4080}\right)\)\(e\left(\frac{2423}{4080}\right)\)\(e\left(\frac{1}{85}\right)\)\(e\left(\frac{331}{510}\right)\)
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)
 
\( \chi_{ 50575 }(1017,a) \;\) at \(\;a = \) e.g. 2