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Properties

Label 1323.625
Modulus $1323$
Conductor $1323$
Order $63$
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(1323, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([14,96]))
 
Copy content gp:[g,chi] = znchar(Mod(625, 1323))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1323.625");
 

Basic properties

Modulus: \(1323\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1323\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(63\)
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 1323.ca

\(\chi_{1323}(25,\cdot)\) \(\chi_{1323}(58,\cdot)\) \(\chi_{1323}(88,\cdot)\) \(\chi_{1323}(121,\cdot)\) \(\chi_{1323}(151,\cdot)\) \(\chi_{1323}(184,\cdot)\) \(\chi_{1323}(247,\cdot)\) \(\chi_{1323}(277,\cdot)\) \(\chi_{1323}(310,\cdot)\) \(\chi_{1323}(340,\cdot)\) \(\chi_{1323}(403,\cdot)\) \(\chi_{1323}(436,\cdot)\) \(\chi_{1323}(466,\cdot)\) \(\chi_{1323}(499,\cdot)\) \(\chi_{1323}(529,\cdot)\) \(\chi_{1323}(562,\cdot)\) \(\chi_{1323}(592,\cdot)\) \(\chi_{1323}(625,\cdot)\) \(\chi_{1323}(688,\cdot)\) \(\chi_{1323}(718,\cdot)\) \(\chi_{1323}(751,\cdot)\) \(\chi_{1323}(781,\cdot)\) \(\chi_{1323}(844,\cdot)\) \(\chi_{1323}(877,\cdot)\) \(\chi_{1323}(907,\cdot)\) \(\chi_{1323}(940,\cdot)\) \(\chi_{1323}(970,\cdot)\) \(\chi_{1323}(1003,\cdot)\) \(\chi_{1323}(1033,\cdot)\) \(\chi_{1323}(1066,\cdot)\) ...

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: $\Q(\zeta_{63})$
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 63 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((785,1081)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{16}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1323 }(625, a) \) \(1\)\(1\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{53}{63}\right)\)\(e\left(\frac{41}{63}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{2}{63}\right)\)\(e\left(\frac{43}{63}\right)\)\(e\left(\frac{5}{7}\right)\)\(1\)
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)
 
\( \chi_{ 1323 }(625,a) \;\) at \(\;a = \) e.g. 2