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Properties

Label 1045.178
Modulus $1045$
Conductor $1045$
Order $60$
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(60)) M = H._module chi = DirichletCharacter(H, M([45,6,20]))
 
Copy content gp:[g,chi] = znchar(Mod(178, 1045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.178");
 

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: \(60\)
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.cf

\(\chi_{1045}(7,\cdot)\) \(\chi_{1045}(68,\cdot)\) \(\chi_{1045}(83,\cdot)\) \(\chi_{1045}(178,\cdot)\) \(\chi_{1045}(182,\cdot)\) \(\chi_{1045}(277,\cdot)\) \(\chi_{1045}(292,\cdot)\) \(\chi_{1045}(387,\cdot)\) \(\chi_{1045}(448,\cdot)\) \(\chi_{1045}(558,\cdot)\) \(\chi_{1045}(657,\cdot)\) \(\chi_{1045}(733,\cdot)\) \(\chi_{1045}(767,\cdot)\) \(\chi_{1045}(843,\cdot)\) \(\chi_{1045}(942,\cdot)\) \(\chi_{1045}(1018,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(178, a) \) \(1\)\(1\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(-i\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{19}{30}\right)\)
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)
 
\( \chi_{ 1045 }(178,a) \;\) at \(\;a = \) e.g. 2