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Properties

Label 3571.87
Modulus $3571$
Conductor $3571$
Order $1190$
Real no
Primitive yes
Minimal yes
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(3571, base_ring=CyclotomicField(1190)) M = H._module chi = DirichletCharacter(H, M([843]))
 
Copy content gp:[g,chi] = znchar(Mod(87, 3571))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3571.87");
 

Basic properties

Modulus: \(3571\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3571\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1190\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3571.bd

\(\chi_{3571}(8,\cdot)\) \(\chi_{3571}(15,\cdot)\) \(\chi_{3571}(23,\cdot)\) \(\chi_{3571}(26,\cdot)\) \(\chi_{3571}(27,\cdot)\) \(\chi_{3571}(35,\cdot)\) \(\chi_{3571}(48,\cdot)\) \(\chi_{3571}(50,\cdot)\) \(\chi_{3571}(57,\cdot)\) \(\chi_{3571}(63,\cdot)\) \(\chi_{3571}(77,\cdot)\) \(\chi_{3571}(82,\cdot)\) \(\chi_{3571}(86,\cdot)\) \(\chi_{3571}(87,\cdot)\) \(\chi_{3571}(110,\cdot)\) \(\chi_{3571}(112,\cdot)\) \(\chi_{3571}(134,\cdot)\) \(\chi_{3571}(136,\cdot)\) \(\chi_{3571}(138,\cdot)\) \(\chi_{3571}(146,\cdot)\) \(\chi_{3571}(147,\cdot)\) \(\chi_{3571}(160,\cdot)\) \(\chi_{3571}(162,\cdot)\) \(\chi_{3571}(194,\cdot)\) \(\chi_{3571}(198,\cdot)\) \(\chi_{3571}(212,\cdot)\) \(\chi_{3571}(222,\cdot)\) \(\chi_{3571}(229,\cdot)\) \(\chi_{3571}(239,\cdot)\) \(\chi_{3571}(244,\cdot)\) ...

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: $\Q(\zeta_{595})$
Fixed field: Number field defined by a degree 1190 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{843}{1190}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3571 }(87, a) \) \(-1\)\(1\)\(e\left(\frac{843}{1190}\right)\)\(e\left(\frac{129}{1190}\right)\)\(e\left(\frac{248}{595}\right)\)\(e\left(\frac{117}{595}\right)\)\(e\left(\frac{486}{595}\right)\)\(e\left(\frac{1189}{1190}\right)\)\(e\left(\frac{149}{1190}\right)\)\(e\left(\frac{129}{595}\right)\)\(e\left(\frac{1077}{1190}\right)\)\(e\left(\frac{386}{595}\right)\)
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)
 
\( \chi_{ 3571 }(87,a) \;\) at \(\;a = \) e.g. 2