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Properties

Label 2419.65
Modulus $2419$
Conductor $2419$
Order $1160$
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(2419, base_ring=CyclotomicField(1160)) M = H._module chi = DirichletCharacter(H, M([377,1020]))
 
Copy content gp:[g,chi] = znchar(Mod(65, 2419))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2419.65");
 

Basic properties

Modulus: \(2419\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2419\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1160\)
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 2419.be

\(\chi_{2419}(6,\cdot)\) \(\chi_{2419}(11,\cdot)\) \(\chi_{2419}(13,\cdot)\) \(\chi_{2419}(24,\cdot)\) \(\chi_{2419}(30,\cdot)\) \(\chi_{2419}(34,\cdot)\) \(\chi_{2419}(47,\cdot)\) \(\chi_{2419}(52,\cdot)\) \(\chi_{2419}(54,\cdot)\) \(\chi_{2419}(56,\cdot)\) \(\chi_{2419}(65,\cdot)\) \(\chi_{2419}(67,\cdot)\) \(\chi_{2419}(69,\cdot)\) \(\chi_{2419}(70,\cdot)\) \(\chi_{2419}(89,\cdot)\) \(\chi_{2419}(93,\cdot)\) \(\chi_{2419}(97,\cdot)\) \(\chi_{2419}(99,\cdot)\) \(\chi_{2419}(101,\cdot)\) \(\chi_{2419}(106,\cdot)\) \(\chi_{2419}(111,\cdot)\) \(\chi_{2419}(129,\cdot)\) \(\chi_{2419}(136,\cdot)\) \(\chi_{2419}(142,\cdot)\) \(\chi_{2419}(149,\cdot)\) \(\chi_{2419}(151,\cdot)\) \(\chi_{2419}(152,\cdot)\) \(\chi_{2419}(157,\cdot)\) \(\chi_{2419}(158,\cdot)\) \(\chi_{2419}(170,\cdot)\) ...

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: $\Q(\zeta_{1160})$
Fixed field: Number field defined by a degree 1160 polynomial (not computed)

Values on generators

\((2302,1477)\) → \((e\left(\frac{13}{40}\right),e\left(\frac{51}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2419 }(65, a) \) \(1\)\(1\)\(e\left(\frac{191}{580}\right)\)\(e\left(\frac{195}{232}\right)\)\(e\left(\frac{191}{290}\right)\)\(e\left(\frac{247}{580}\right)\)\(e\left(\frac{197}{1160}\right)\)\(e\left(\frac{583}{1160}\right)\)\(e\left(\frac{573}{580}\right)\)\(e\left(\frac{79}{116}\right)\)\(e\left(\frac{219}{290}\right)\)\(e\left(\frac{1111}{1160}\right)\)
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)
 
\( \chi_{ 2419 }(65,a) \;\) at \(\;a = \) e.g. 2