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Properties

Label 1751.48
Modulus $1751$
Conductor $1751$
Order $816$
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(1751, base_ring=CyclotomicField(816)) M = H._module chi = DirichletCharacter(H, M([459,88]))
 
Copy content gp:[g,chi] = znchar(Mod(48, 1751))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1751.48");
 

Basic properties

Modulus: \(1751\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1751\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(816\)
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 1751.bm

\(\chi_{1751}(5,\cdot)\) \(\chi_{1751}(6,\cdot)\) \(\chi_{1751}(11,\cdot)\) \(\chi_{1751}(12,\cdot)\) \(\chi_{1751}(20,\cdot)\) \(\chi_{1751}(40,\cdot)\) \(\chi_{1751}(44,\cdot)\) \(\chi_{1751}(45,\cdot)\) \(\chi_{1751}(48,\cdot)\) \(\chi_{1751}(54,\cdot)\) \(\chi_{1751}(62,\cdot)\) \(\chi_{1751}(65,\cdot)\) \(\chi_{1751}(71,\cdot)\) \(\chi_{1751}(74,\cdot)\) \(\chi_{1751}(75,\cdot)\) \(\chi_{1751}(78,\cdot)\) \(\chi_{1751}(88,\cdot)\) \(\chi_{1751}(96,\cdot)\) \(\chi_{1751}(99,\cdot)\) \(\chi_{1751}(108,\cdot)\) \(\chi_{1751}(109,\cdot)\) \(\chi_{1751}(114,\cdot)\) \(\chi_{1751}(124,\cdot)\) \(\chi_{1751}(143,\cdot)\) \(\chi_{1751}(146,\cdot)\) \(\chi_{1751}(147,\cdot)\) \(\chi_{1751}(148,\cdot)\) \(\chi_{1751}(156,\cdot)\) \(\chi_{1751}(165,\cdot)\) \(\chi_{1751}(173,\cdot)\) ...

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: $\Q(\zeta_{816})$
Fixed field: Number field defined by a degree 816 polynomial (not computed)

Values on generators

\((207,1344)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{11}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1751 }(48, a) \) \(1\)\(1\)\(e\left(\frac{253}{408}\right)\)\(e\left(\frac{209}{272}\right)\)\(e\left(\frac{49}{204}\right)\)\(e\left(\frac{751}{816}\right)\)\(e\left(\frac{317}{816}\right)\)\(e\left(\frac{505}{816}\right)\)\(e\left(\frac{117}{136}\right)\)\(e\left(\frac{73}{136}\right)\)\(e\left(\frac{147}{272}\right)\)\(e\left(\frac{421}{816}\right)\)
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)
 
\( \chi_{ 1751 }(48,a) \;\) at \(\;a = \) e.g. 2