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Properties

Label 1323.835
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([70,78]))
 
Copy content gp:[g,chi] = znchar(Mod(835, 1323))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1323.835");
 

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.cc

\(\chi_{1323}(4,\cdot)\) \(\chi_{1323}(16,\cdot)\) \(\chi_{1323}(130,\cdot)\) \(\chi_{1323}(142,\cdot)\) \(\chi_{1323}(193,\cdot)\) \(\chi_{1323}(205,\cdot)\) \(\chi_{1323}(256,\cdot)\) \(\chi_{1323}(268,\cdot)\) \(\chi_{1323}(319,\cdot)\) \(\chi_{1323}(331,\cdot)\) \(\chi_{1323}(382,\cdot)\) \(\chi_{1323}(394,\cdot)\) \(\chi_{1323}(445,\cdot)\) \(\chi_{1323}(457,\cdot)\) \(\chi_{1323}(571,\cdot)\) \(\chi_{1323}(583,\cdot)\) \(\chi_{1323}(634,\cdot)\) \(\chi_{1323}(646,\cdot)\) \(\chi_{1323}(697,\cdot)\) \(\chi_{1323}(709,\cdot)\) \(\chi_{1323}(760,\cdot)\) \(\chi_{1323}(772,\cdot)\) \(\chi_{1323}(823,\cdot)\) \(\chi_{1323}(835,\cdot)\) \(\chi_{1323}(886,\cdot)\) \(\chi_{1323}(898,\cdot)\) \(\chi_{1323}(1012,\cdot)\) \(\chi_{1323}(1024,\cdot)\) \(\chi_{1323}(1075,\cdot)\) \(\chi_{1323}(1087,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{13}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1323 }(835, a) \) \(1\)\(1\)\(e\left(\frac{41}{63}\right)\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{62}{63}\right)\)\(e\left(\frac{55}{63}\right)\)\(e\left(\frac{38}{63}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{3}\right)\)
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)
 
\( \chi_{ 1323 }(835,a) \;\) at \(\;a = \) e.g. 2