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Properties

Label 1373.87
Modulus $1373$
Conductor $1373$
Order $686$
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(1373, base_ring=CyclotomicField(686)) M = H._module chi = DirichletCharacter(H, M([43]))
 
Copy content gp:[g,chi] = znchar(Mod(87, 1373))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1373.87");
 

Basic properties

Modulus: \(1373\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1373\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(686\)
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 1373.k

\(\chi_{1373}(4,\cdot)\) \(\chi_{1373}(9,\cdot)\) \(\chi_{1373}(10,\cdot)\) \(\chi_{1373}(11,\cdot)\) \(\chi_{1373}(17,\cdot)\) \(\chi_{1373}(19,\cdot)\) \(\chi_{1373}(23,\cdot)\) \(\chi_{1373}(24,\cdot)\) \(\chi_{1373}(25,\cdot)\) \(\chi_{1373}(26,\cdot)\) \(\chi_{1373}(28,\cdot)\) \(\chi_{1373}(37,\cdot)\) \(\chi_{1373}(43,\cdot)\) \(\chi_{1373}(54,\cdot)\) \(\chi_{1373}(60,\cdot)\) \(\chi_{1373}(63,\cdot)\) \(\chi_{1373}(64,\cdot)\) \(\chi_{1373}(65,\cdot)\) \(\chi_{1373}(66,\cdot)\) \(\chi_{1373}(70,\cdot)\) \(\chi_{1373}(77,\cdot)\) \(\chi_{1373}(87,\cdot)\) \(\chi_{1373}(94,\cdot)\) \(\chi_{1373}(102,\cdot)\) \(\chi_{1373}(107,\cdot)\) \(\chi_{1373}(114,\cdot)\) \(\chi_{1373}(119,\cdot)\) \(\chi_{1373}(124,\cdot)\) \(\chi_{1373}(131,\cdot)\) \(\chi_{1373}(133,\cdot)\) ...

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: $\Q(\zeta_{343})$
Fixed field: Number field defined by a degree 686 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{43}{686}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1373 }(87, a) \) \(1\)\(1\)\(e\left(\frac{43}{686}\right)\)\(e\left(\frac{591}{686}\right)\)\(e\left(\frac{43}{343}\right)\)\(e\left(\frac{379}{686}\right)\)\(e\left(\frac{317}{343}\right)\)\(e\left(\frac{100}{343}\right)\)\(e\left(\frac{129}{686}\right)\)\(e\left(\frac{248}{343}\right)\)\(e\left(\frac{211}{343}\right)\)\(e\left(\frac{139}{343}\right)\)
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)
 
\( \chi_{ 1373 }(87,a) \;\) at \(\;a = \) e.g. 2