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Properties

Label 1477.1109
Modulus $1477$
Conductor $1477$
Order $42$
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(1477, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([7,26]))
 
Copy content gp:[g,chi] = znchar(Mod(1109, 1477))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1477.1109");
 

Basic properties

Modulus: \(1477\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1477\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42\)
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 1477.cg

\(\chi_{1477}(101,\cdot)\) \(\chi_{1477}(117,\cdot)\) \(\chi_{1477}(173,\cdot)\) \(\chi_{1477}(178,\cdot)\) \(\chi_{1477}(185,\cdot)\) \(\chi_{1477}(390,\cdot)\) \(\chi_{1477}(495,\cdot)\) \(\chi_{1477}(794,\cdot)\) \(\chi_{1477}(887,\cdot)\) \(\chi_{1477}(1109,\cdot)\) \(\chi_{1477}(1235,\cdot)\) \(\chi_{1477}(1300,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((423,1268)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1477 }(1109, a) \) \(-1\)\(1\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(-1\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{29}{42}\right)\)
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)
 
\( \chi_{ 1477 }(1109,a) \;\) at \(\;a = \) e.g. 2