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Properties

Label 1613.704
Modulus $1613$
Conductor $1613$
Order $124$
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(1613, base_ring=CyclotomicField(124)) M = H._module chi = DirichletCharacter(H, M([5]))
 
Copy content gp:[g,chi] = znchar(Mod(704, 1613))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1613.704");
 

Basic properties

Modulus: \(1613\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1613\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(124\)
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 1613.i

\(\chi_{1613}(80,\cdot)\) \(\chi_{1613}(152,\cdot)\) \(\chi_{1613}(161,\cdot)\) \(\chi_{1613}(165,\cdot)\) \(\chi_{1613}(166,\cdot)\) \(\chi_{1613}(172,\cdot)\) \(\chi_{1613}(178,\cdot)\) \(\chi_{1613}(276,\cdot)\) \(\chi_{1613}(307,\cdot)\) \(\chi_{1613}(422,\cdot)\) \(\chi_{1613}(449,\cdot)\) \(\chi_{1613}(450,\cdot)\) \(\chi_{1613}(491,\cdot)\) \(\chi_{1613}(493,\cdot)\) \(\chi_{1613}(515,\cdot)\) \(\chi_{1613}(541,\cdot)\) \(\chi_{1613}(544,\cdot)\) \(\chi_{1613}(567,\cdot)\) \(\chi_{1613}(598,\cdot)\) \(\chi_{1613}(638,\cdot)\) \(\chi_{1613}(641,\cdot)\) \(\chi_{1613}(679,\cdot)\) \(\chi_{1613}(697,\cdot)\) \(\chi_{1613}(704,\cdot)\) \(\chi_{1613}(711,\cdot)\) \(\chi_{1613}(734,\cdot)\) \(\chi_{1613}(746,\cdot)\) \(\chi_{1613}(758,\cdot)\) \(\chi_{1613}(766,\cdot)\) \(\chi_{1613}(795,\cdot)\) ...

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: $\Q(\zeta_{124})$
Fixed field: Number field defined by a degree 124 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{5}{124}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1613 }(704, a) \) \(-1\)\(1\)\(-i\)\(e\left(\frac{5}{124}\right)\)\(-1\)\(e\left(\frac{63}{124}\right)\)\(e\left(\frac{49}{62}\right)\)\(e\left(\frac{21}{124}\right)\)\(i\)\(e\left(\frac{5}{62}\right)\)\(e\left(\frac{8}{31}\right)\)\(e\left(\frac{15}{124}\right)\)
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)
 
\( \chi_{ 1613 }(704,a) \;\) at \(\;a = \) e.g. 2