Properties

Label 1045.784
Modulus $1045$
Conductor $1045$
Order $90$
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(1045, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([45,72,80]))
 
Copy content gp:[g,chi] = znchar(Mod(784, 1045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.784");
 

Basic properties

Modulus: \(1045\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1045\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 1045.ck

\(\chi_{1045}(4,\cdot)\) \(\chi_{1045}(9,\cdot)\) \(\chi_{1045}(104,\cdot)\) \(\chi_{1045}(119,\cdot)\) \(\chi_{1045}(169,\cdot)\) \(\chi_{1045}(214,\cdot)\) \(\chi_{1045}(234,\cdot)\) \(\chi_{1045}(289,\cdot)\) \(\chi_{1045}(339,\cdot)\) \(\chi_{1045}(389,\cdot)\) \(\chi_{1045}(434,\cdot)\) \(\chi_{1045}(454,\cdot)\) \(\chi_{1045}(499,\cdot)\) \(\chi_{1045}(614,\cdot)\) \(\chi_{1045}(669,\cdot)\) \(\chi_{1045}(674,\cdot)\) \(\chi_{1045}(709,\cdot)\) \(\chi_{1045}(719,\cdot)\) \(\chi_{1045}(764,\cdot)\) \(\chi_{1045}(784,\cdot)\) \(\chi_{1045}(834,\cdot)\) \(\chi_{1045}(929,\cdot)\) \(\chi_{1045}(994,\cdot)\) \(\chi_{1045}(1004,\cdot)\)

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: $\Q(\zeta_{45})$
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 90 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((837,761,496)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(784, a) \) \(1\)\(1\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{28}{45}\right)\)
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)
 
\( \chi_{ 1045 }(784,a) \;\) at \(\;a = \) e.g. 2