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Properties

Label 1645.257
Modulus $1645$
Conductor $1645$
Order $276$
Real no
Primitive yes
Minimal yes
Parity odd

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(1645, base_ring=CyclotomicField(276)) M = H._module chi = DirichletCharacter(H, M([69,230,150]))
 
Copy content gp:[g,chi] = znchar(Mod(257, 1645))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1645.257");
 

Basic properties

Modulus: \(1645\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1645\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(276\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1645.bu

\(\chi_{1645}(33,\cdot)\) \(\chi_{1645}(38,\cdot)\) \(\chi_{1645}(52,\cdot)\) \(\chi_{1645}(73,\cdot)\) \(\chi_{1645}(82,\cdot)\) \(\chi_{1645}(87,\cdot)\) \(\chi_{1645}(117,\cdot)\) \(\chi_{1645}(138,\cdot)\) \(\chi_{1645}(152,\cdot)\) \(\chi_{1645}(208,\cdot)\) \(\chi_{1645}(227,\cdot)\) \(\chi_{1645}(248,\cdot)\) \(\chi_{1645}(257,\cdot)\) \(\chi_{1645}(278,\cdot)\) \(\chi_{1645}(292,\cdot)\) \(\chi_{1645}(297,\cdot)\) \(\chi_{1645}(313,\cdot)\) \(\chi_{1645}(327,\cdot)\) \(\chi_{1645}(348,\cdot)\) \(\chi_{1645}(362,\cdot)\) \(\chi_{1645}(367,\cdot)\) \(\chi_{1645}(402,\cdot)\) \(\chi_{1645}(453,\cdot)\) \(\chi_{1645}(458,\cdot)\) \(\chi_{1645}(467,\cdot)\) \(\chi_{1645}(493,\cdot)\) \(\chi_{1645}(528,\cdot)\) \(\chi_{1645}(537,\cdot)\) \(\chi_{1645}(558,\cdot)\) \(\chi_{1645}(577,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((1317,941,1086)\) → \((i,e\left(\frac{5}{6}\right),e\left(\frac{25}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1645 }(257, a) \) \(-1\)\(1\)\(e\left(\frac{193}{276}\right)\)\(e\left(\frac{125}{276}\right)\)\(e\left(\frac{55}{138}\right)\)\(e\left(\frac{7}{46}\right)\)\(e\left(\frac{9}{92}\right)\)\(e\left(\frac{125}{138}\right)\)\(e\left(\frac{19}{138}\right)\)\(e\left(\frac{235}{276}\right)\)\(e\left(\frac{21}{92}\right)\)\(e\left(\frac{55}{69}\right)\)
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)
 
\( \chi_{ 1645 }(257,a) \;\) at \(\;a = \) e.g. 2