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Properties

Label 2401.13
Modulus $2401$
Conductor $2401$
Order $686$
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(2401, base_ring=CyclotomicField(686)) M = H._module chi = DirichletCharacter(H, M([291]))
 
Copy content gp:[g,chi] = znchar(Mod(13, 2401))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.13");
 

Basic properties

Modulus: \(2401\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2401\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2401.n

\(\chi_{2401}(6,\cdot)\) \(\chi_{2401}(13,\cdot)\) \(\chi_{2401}(20,\cdot)\) \(\chi_{2401}(27,\cdot)\) \(\chi_{2401}(34,\cdot)\) \(\chi_{2401}(41,\cdot)\) \(\chi_{2401}(55,\cdot)\) \(\chi_{2401}(62,\cdot)\) \(\chi_{2401}(69,\cdot)\) \(\chi_{2401}(76,\cdot)\) \(\chi_{2401}(83,\cdot)\) \(\chi_{2401}(90,\cdot)\) \(\chi_{2401}(104,\cdot)\) \(\chi_{2401}(111,\cdot)\) \(\chi_{2401}(118,\cdot)\) \(\chi_{2401}(125,\cdot)\) \(\chi_{2401}(132,\cdot)\) \(\chi_{2401}(139,\cdot)\) \(\chi_{2401}(153,\cdot)\) \(\chi_{2401}(160,\cdot)\) \(\chi_{2401}(167,\cdot)\) \(\chi_{2401}(174,\cdot)\) \(\chi_{2401}(181,\cdot)\) \(\chi_{2401}(188,\cdot)\) \(\chi_{2401}(202,\cdot)\) \(\chi_{2401}(209,\cdot)\) \(\chi_{2401}(216,\cdot)\) \(\chi_{2401}(223,\cdot)\) \(\chi_{2401}(230,\cdot)\) \(\chi_{2401}(237,\cdot)\) ...

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: $\Q(\zeta_{343})$
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 686 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(3\) → \(e\left(\frac{291}{686}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2401 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{150}{343}\right)\)\(e\left(\frac{291}{686}\right)\)\(e\left(\frac{300}{343}\right)\)\(e\left(\frac{305}{686}\right)\)\(e\left(\frac{591}{686}\right)\)\(e\left(\frac{107}{343}\right)\)\(e\left(\frac{291}{343}\right)\)\(e\left(\frac{605}{686}\right)\)\(e\left(\frac{276}{343}\right)\)\(e\left(\frac{205}{686}\right)\)
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)
 
\( \chi_{ 2401 }(13,a) \;\) at \(\;a = \) e.g. 2