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Properties

Label 3571.99
Modulus $3571$
Conductor $3571$
Order $357$
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(3571, base_ring=CyclotomicField(714)) M = H._module chi = DirichletCharacter(H, M([568]))
 
Copy content gp:[g,chi] = znchar(Mod(99, 3571))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3571.99");
 

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: \(357\)
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 3571.z

\(\chi_{3571}(43,\cdot)\) \(\chi_{3571}(54,\cdot)\) \(\chi_{3571}(67,\cdot)\) \(\chi_{3571}(99,\cdot)\) \(\chi_{3571}(124,\cdot)\) \(\chi_{3571}(126,\cdot)\) \(\chi_{3571}(144,\cdot)\) \(\chi_{3571}(150,\cdot)\) \(\chi_{3571}(164,\cdot)\) \(\chi_{3571}(174,\cdot)\) \(\chi_{3571}(214,\cdot)\) \(\chi_{3571}(226,\cdot)\) \(\chi_{3571}(231,\cdot)\) \(\chi_{3571}(271,\cdot)\) \(\chi_{3571}(277,\cdot)\) \(\chi_{3571}(294,\cdot)\) \(\chi_{3571}(319,\cdot)\) \(\chi_{3571}(350,\cdot)\) \(\chi_{3571}(356,\cdot)\) \(\chi_{3571}(403,\cdot)\) \(\chi_{3571}(406,\cdot)\) \(\chi_{3571}(414,\cdot)\) \(\chi_{3571}(417,\cdot)\) \(\chi_{3571}(444,\cdot)\) \(\chi_{3571}(464,\cdot)\) \(\chi_{3571}(468,\cdot)\) \(\chi_{3571}(498,\cdot)\) \(\chi_{3571}(505,\cdot)\) \(\chi_{3571}(533,\cdot)\) \(\chi_{3571}(566,\cdot)\) ...

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: $\Q(\zeta_{357})$
Fixed field: Number field defined by a degree 357 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{284}{357}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3571 }(99, a) \) \(1\)\(1\)\(e\left(\frac{284}{357}\right)\)\(e\left(\frac{46}{357}\right)\)\(e\left(\frac{211}{357}\right)\)\(e\left(\frac{53}{357}\right)\)\(e\left(\frac{110}{119}\right)\)\(e\left(\frac{43}{357}\right)\)\(e\left(\frac{46}{119}\right)\)\(e\left(\frac{92}{357}\right)\)\(e\left(\frac{337}{357}\right)\)\(e\left(\frac{5}{357}\right)\)
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)
 
\( \chi_{ 3571 }(99,a) \;\) at \(\;a = \) e.g. 2