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Properties

Label 1045.459
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([45,27,65]))
 
Copy content gp:[g,chi] = znchar(Mod(459, 1045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.459");
 

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.cn

\(\chi_{1045}(29,\cdot)\) \(\chi_{1045}(79,\cdot)\) \(\chi_{1045}(129,\cdot)\) \(\chi_{1045}(184,\cdot)\) \(\chi_{1045}(204,\cdot)\) \(\chi_{1045}(249,\cdot)\) \(\chi_{1045}(299,\cdot)\) \(\chi_{1045}(314,\cdot)\) \(\chi_{1045}(409,\cdot)\) \(\chi_{1045}(414,\cdot)\) \(\chi_{1045}(459,\cdot)\) \(\chi_{1045}(469,\cdot)\) \(\chi_{1045}(534,\cdot)\) \(\chi_{1045}(629,\cdot)\) \(\chi_{1045}(679,\cdot)\) \(\chi_{1045}(699,\cdot)\) \(\chi_{1045}(744,\cdot)\) \(\chi_{1045}(754,\cdot)\) \(\chi_{1045}(789,\cdot)\) \(\chi_{1045}(794,\cdot)\) \(\chi_{1045}(849,\cdot)\) \(\chi_{1045}(964,\cdot)\) \(\chi_{1045}(1009,\cdot)\) \(\chi_{1045}(1029,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((837,761,496)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(459, a) \) \(1\)\(1\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{41}{90}\right)\)
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)
 
\( \chi_{ 1045 }(459,a) \;\) at \(\;a = \) e.g. 2