↑

Properties

Label 1571.143
Modulus $1571$
Conductor $1571$
Order $785$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Error: 2.2.1.d3936600 is not the label of a Sato-Tate group currently in the database.

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1571, base_ring=CyclotomicField(1570)) M = H._module chi = DirichletCharacter(H, M([148]))
 
Copy content gp:[g,chi] = znchar(Mod(143, 1571))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1571.143");
 

Basic properties

Modulus: \(1571\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1571\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(785\)
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 1571.g

\(\chi_{1571}(3,\cdot)\) \(\chi_{1571}(4,\cdot)\) \(\chi_{1571}(5,\cdot)\) \(\chi_{1571}(7,\cdot)\) \(\chi_{1571}(9,\cdot)\) \(\chi_{1571}(12,\cdot)\) \(\chi_{1571}(16,\cdot)\) \(\chi_{1571}(19,\cdot)\) \(\chi_{1571}(20,\cdot)\) \(\chi_{1571}(21,\cdot)\) \(\chi_{1571}(22,\cdot)\) \(\chi_{1571}(25,\cdot)\) \(\chi_{1571}(26,\cdot)\) \(\chi_{1571}(27,\cdot)\) \(\chi_{1571}(28,\cdot)\) \(\chi_{1571}(29,\cdot)\) \(\chi_{1571}(31,\cdot)\) \(\chi_{1571}(34,\cdot)\) \(\chi_{1571}(35,\cdot)\) \(\chi_{1571}(45,\cdot)\) \(\chi_{1571}(47,\cdot)\) \(\chi_{1571}(48,\cdot)\) \(\chi_{1571}(49,\cdot)\) \(\chi_{1571}(57,\cdot)\) \(\chi_{1571}(59,\cdot)\) \(\chi_{1571}(60,\cdot)\) \(\chi_{1571}(61,\cdot)\) \(\chi_{1571}(64,\cdot)\) \(\chi_{1571}(66,\cdot)\) \(\chi_{1571}(67,\cdot)\) ...

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: $\Q(\zeta_{785})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 785 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(2\) → \(e\left(\frac{74}{785}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1571 }(143, a) \) \(1\)\(1\)\(e\left(\frac{74}{785}\right)\)\(e\left(\frac{506}{785}\right)\)\(e\left(\frac{148}{785}\right)\)\(e\left(\frac{319}{785}\right)\)\(e\left(\frac{116}{157}\right)\)\(e\left(\frac{68}{785}\right)\)\(e\left(\frac{222}{785}\right)\)\(e\left(\frac{227}{785}\right)\)\(e\left(\frac{393}{785}\right)\)\(e\left(\frac{112}{785}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1571 }(143,a) \;\) at \(\;a = \) e.g. 2