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Properties

Label 4361.1014
Modulus $4361$
Conductor $4361$
Order $616$
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(4361, base_ring=CyclotomicField(616)) M = H._module chi = DirichletCharacter(H, M([132,441]))
 
Copy content gp:[g,chi] = znchar(Mod(1014, 4361))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4361.1014");
 

Basic properties

Modulus: \(4361\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4361\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(616\)
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 4361.cg

\(\chi_{4361}(6,\cdot)\) \(\chi_{4361}(13,\cdot)\) \(\chi_{4361}(27,\cdot)\) \(\chi_{4361}(41,\cdot)\) \(\chi_{4361}(62,\cdot)\) \(\chi_{4361}(76,\cdot)\) \(\chi_{4361}(83,\cdot)\) \(\chi_{4361}(104,\cdot)\) \(\chi_{4361}(118,\cdot)\) \(\chi_{4361}(132,\cdot)\) \(\chi_{4361}(181,\cdot)\) \(\chi_{4361}(202,\cdot)\) \(\chi_{4361}(209,\cdot)\) \(\chi_{4361}(216,\cdot)\) \(\chi_{4361}(237,\cdot)\) \(\chi_{4361}(286,\cdot)\) \(\chi_{4361}(300,\cdot)\) \(\chi_{4361}(321,\cdot)\) \(\chi_{4361}(328,\cdot)\) \(\chi_{4361}(349,\cdot)\) \(\chi_{4361}(363,\cdot)\) \(\chi_{4361}(370,\cdot)\) \(\chi_{4361}(384,\cdot)\) \(\chi_{4361}(412,\cdot)\) \(\chi_{4361}(419,\cdot)\) \(\chi_{4361}(426,\cdot)\) \(\chi_{4361}(468,\cdot)\) \(\chi_{4361}(475,\cdot)\) \(\chi_{4361}(496,\cdot)\) \(\chi_{4361}(503,\cdot)\) ...

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: $\Q(\zeta_{616})$
Fixed field: Number field defined by a degree 616 polynomial (not computed)

Values on generators

\((2404,1961)\) → \((e\left(\frac{3}{14}\right),e\left(\frac{63}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 4361 }(1014, a) \) \(1\)\(1\)\(e\left(\frac{2}{77}\right)\)\(e\left(\frac{573}{616}\right)\)\(e\left(\frac{4}{77}\right)\)\(e\left(\frac{101}{308}\right)\)\(e\left(\frac{589}{616}\right)\)\(e\left(\frac{6}{77}\right)\)\(e\left(\frac{265}{308}\right)\)\(e\left(\frac{109}{308}\right)\)\(e\left(\frac{109}{154}\right)\)\(e\left(\frac{55}{56}\right)\)
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)
 
\( \chi_{ 4361 }(1014,a) \;\) at \(\;a = \) e.g. 2