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Properties

Label 51264.16583
Modulus $51264$
Conductor $25632$
Order $264$
Real no
Primitive no
Minimal no
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(51264, base_ring=CyclotomicField(264)) M = H._module chi = DirichletCharacter(H, M([132,165,220,177]))
 
Copy content gp:[g,chi] = znchar(Mod(16583, 51264))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("51264.16583");
 

Basic properties

Modulus: \(51264\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(25632\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(264\)
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: no, induced from \(\chi_{25632}(19787,\cdot)\)
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: no
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 51264.qj

\(\chi_{51264}(23,\cdot)\) \(\chi_{51264}(743,\cdot)\) \(\chi_{51264}(839,\cdot)\) \(\chi_{51264}(887,\cdot)\) \(\chi_{51264}(1559,\cdot)\) \(\chi_{51264}(1895,\cdot)\) \(\chi_{51264}(4631,\cdot)\) \(\chi_{51264}(5495,\cdot)\) \(\chi_{51264}(6215,\cdot)\) \(\chi_{51264}(6647,\cdot)\) \(\chi_{51264}(6791,\cdot)\) \(\chi_{51264}(9671,\cdot)\) \(\chi_{51264}(10775,\cdot)\) \(\chi_{51264}(10823,\cdot)\) \(\chi_{51264}(11255,\cdot)\) \(\chi_{51264}(11351,\cdot)\) \(\chi_{51264}(11831,\cdot)\) \(\chi_{51264}(12263,\cdot)\) \(\chi_{51264}(13223,\cdot)\) \(\chi_{51264}(14567,\cdot)\) \(\chi_{51264}(15143,\cdot)\) \(\chi_{51264}(15959,\cdot)\) \(\chi_{51264}(16007,\cdot)\) \(\chi_{51264}(16583,\cdot)\) \(\chi_{51264}(17111,\cdot)\) \(\chi_{51264}(17831,\cdot)\) \(\chi_{51264}(17975,\cdot)\) \(\chi_{51264}(18887,\cdot)\) \(\chi_{51264}(18983,\cdot)\) \(\chi_{51264}(20327,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((49663,3205,45569,9793)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{59}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 51264 }(16583, a) \) \(-1\)\(1\)\(e\left(\frac{191}{264}\right)\)\(e\left(\frac{103}{264}\right)\)\(e\left(\frac{205}{264}\right)\)\(e\left(\frac{61}{132}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{167}{264}\right)\)\(e\left(\frac{59}{132}\right)\)\(e\left(\frac{35}{132}\right)\)\(e\left(\frac{251}{264}\right)\)
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)
 
\( \chi_{ 51264 }(16583,a) \;\) at \(\;a = \) e.g. 2