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Properties

Label 6556.1153
Modulus $6556$
Conductor $1639$
Order $370$
Real no
Primitive no
Minimal yes
Parity even

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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(6556, base_ring=CyclotomicField(370)) M = H._module chi = DirichletCharacter(H, M([0,222,165]))
 
Copy content gp:[g,chi] = znchar(Mod(1153, 6556))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6556.1153");
 

Basic properties

Modulus: \(6556\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1639\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(370\)
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: no, induced from \(\chi_{1639}(1153,\cdot)\)
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 6556.bo

\(\chi_{6556}(9,\cdot)\) \(\chi_{6556}(53,\cdot)\) \(\chi_{6556}(69,\cdot)\) \(\chi_{6556}(113,\cdot)\) \(\chi_{6556}(169,\cdot)\) \(\chi_{6556}(213,\cdot)\) \(\chi_{6556}(225,\cdot)\) \(\chi_{6556}(269,\cdot)\) \(\chi_{6556}(273,\cdot)\) \(\chi_{6556}(333,\cdot)\) \(\chi_{6556}(345,\cdot)\) \(\chi_{6556}(401,\cdot)\) \(\chi_{6556}(489,\cdot)\) \(\chi_{6556}(533,\cdot)\) \(\chi_{6556}(565,\cdot)\) \(\chi_{6556}(577,\cdot)\) \(\chi_{6556}(641,\cdot)\) \(\chi_{6556}(665,\cdot)\) \(\chi_{6556}(709,\cdot)\) \(\chi_{6556}(729,\cdot)\) \(\chi_{6556}(845,\cdot)\) \(\chi_{6556}(861,\cdot)\) \(\chi_{6556}(889,\cdot)\) \(\chi_{6556}(929,\cdot)\) \(\chi_{6556}(1037,\cdot)\) \(\chi_{6556}(1065,\cdot)\) \(\chi_{6556}(1125,\cdot)\) \(\chi_{6556}(1153,\cdot)\) \(\chi_{6556}(1237,\cdot)\) \(\chi_{6556}(1313,\cdot)\) ...

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: $\Q(\zeta_{185})$
Fixed field: Number field defined by a degree 370 polynomial (not computed)

Values on generators

\((3279,3577,4621)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{33}{74}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 6556 }(1153, a) \) \(1\)\(1\)\(e\left(\frac{221}{370}\right)\)\(e\left(\frac{144}{185}\right)\)\(e\left(\frac{97}{185}\right)\)\(e\left(\frac{36}{185}\right)\)\(e\left(\frac{87}{370}\right)\)\(e\left(\frac{139}{370}\right)\)\(e\left(\frac{129}{185}\right)\)\(e\left(\frac{48}{185}\right)\)\(e\left(\frac{9}{74}\right)\)\(e\left(\frac{27}{74}\right)\)
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)
 
\( \chi_{ 6556 }(1153,a) \;\) at \(\;a = \) e.g. 2