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Properties

Label 1045.333
Modulus $1045$
Conductor $1045$
Order $180$
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(180)) M = H._module chi = DirichletCharacter(H, M([135,144,170]))
 
Copy content gp:[g,chi] = znchar(Mod(333, 1045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.333");
 

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: \(180\)
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.cr

\(\chi_{1045}(3,\cdot)\) \(\chi_{1045}(48,\cdot)\) \(\chi_{1045}(53,\cdot)\) \(\chi_{1045}(97,\cdot)\) \(\chi_{1045}(108,\cdot)\) \(\chi_{1045}(147,\cdot)\) \(\chi_{1045}(148,\cdot)\) \(\chi_{1045}(192,\cdot)\) \(\chi_{1045}(203,\cdot)\) \(\chi_{1045}(212,\cdot)\) \(\chi_{1045}(223,\cdot)\) \(\chi_{1045}(257,\cdot)\) \(\chi_{1045}(262,\cdot)\) \(\chi_{1045}(268,\cdot)\) \(\chi_{1045}(317,\cdot)\) \(\chi_{1045}(333,\cdot)\) \(\chi_{1045}(357,\cdot)\) \(\chi_{1045}(383,\cdot)\) \(\chi_{1045}(412,\cdot)\) \(\chi_{1045}(432,\cdot)\) \(\chi_{1045}(433,\cdot)\) \(\chi_{1045}(477,\cdot)\) \(\chi_{1045}(478,\cdot)\) \(\chi_{1045}(488,\cdot)\) \(\chi_{1045}(542,\cdot)\) \(\chi_{1045}(553,\cdot)\) \(\chi_{1045}(592,\cdot)\) \(\chi_{1045}(603,\cdot)\) \(\chi_{1045}(642,\cdot)\) \(\chi_{1045}(687,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((837,761,496)\) → \((-i,e\left(\frac{4}{5}\right),e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(333, a) \) \(1\)\(1\)\(e\left(\frac{89}{180}\right)\)\(e\left(\frac{167}{180}\right)\)\(e\left(\frac{89}{90}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{139}{180}\right)\)\(e\left(\frac{23}{45}\right)\)
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)
 
\( \chi_{ 1045 }(333,a) \;\) at \(\;a = \) e.g. 2