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Properties

Label 512000.10743
Modulus $512000$
Conductor $51200$
Order $2560$
Real no
Primitive no
Minimal no
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(512000, base_ring=CyclotomicField(2560)) M = H._module chi = DirichletCharacter(H, M([1280,2415,896]))
 
Copy content gp:[g,chi] = znchar(Mod(10743, 512000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("512000.10743");
 

Basic properties

Modulus: \(512000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(51200\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2560\)
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 \(\chi_{51200}(6403,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 512000.ic

\(\chi_{512000}(7,\cdot)\) \(\chi_{512000}(343,\cdot)\) \(\chi_{512000}(1143,\cdot)\) \(\chi_{512000}(1607,\cdot)\) \(\chi_{512000}(2407,\cdot)\) \(\chi_{512000}(2743,\cdot)\) \(\chi_{512000}(3207,\cdot)\) \(\chi_{512000}(3543,\cdot)\) \(\chi_{512000}(4007,\cdot)\) \(\chi_{512000}(4343,\cdot)\) \(\chi_{512000}(5143,\cdot)\) \(\chi_{512000}(5607,\cdot)\) \(\chi_{512000}(6407,\cdot)\) \(\chi_{512000}(6743,\cdot)\) \(\chi_{512000}(7207,\cdot)\) \(\chi_{512000}(7543,\cdot)\) \(\chi_{512000}(8007,\cdot)\) \(\chi_{512000}(8343,\cdot)\) \(\chi_{512000}(9143,\cdot)\) \(\chi_{512000}(9607,\cdot)\) \(\chi_{512000}(10407,\cdot)\) \(\chi_{512000}(10743,\cdot)\) \(\chi_{512000}(11207,\cdot)\) \(\chi_{512000}(11543,\cdot)\) \(\chi_{512000}(12007,\cdot)\) \(\chi_{512000}(12343,\cdot)\) \(\chi_{512000}(13143,\cdot)\) \(\chi_{512000}(13607,\cdot)\) \(\chi_{512000}(14407,\cdot)\) \(\chi_{512000}(14743,\cdot)\) ...

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: $\Q(\zeta_{2560})$
Fixed field: Number field defined by a degree 2560 polynomial (not computed)

Values on generators

\((458751,106501,229377)\) → \((-1,e\left(\frac{483}{512}\right),e\left(\frac{7}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 512000 }(10743, a) \) \(1\)\(1\)\(e\left(\frac{1837}{2560}\right)\)\(e\left(\frac{207}{256}\right)\)\(e\left(\frac{557}{1280}\right)\)\(e\left(\frac{91}{2560}\right)\)\(e\left(\frac{1569}{2560}\right)\)\(e\left(\frac{137}{640}\right)\)\(e\left(\frac{1913}{2560}\right)\)\(e\left(\frac{1347}{2560}\right)\)\(e\left(\frac{393}{1280}\right)\)\(e\left(\frac{391}{2560}\right)\)
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)
 
\( \chi_{ 512000 }(10743,a) \;\) at \(\;a = \) e.g. 2