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Properties

Label 6545.1698
Modulus $6545$
Conductor $6545$
Order $120$
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(6545, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([90,80,24,45]))
 
Copy content gp:[g,chi] = znchar(Mod(1698, 6545))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6545.1698");
 

Basic properties

Modulus: \(6545\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6545\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(120\)
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 6545.kj

\(\chi_{6545}(53,\cdot)\) \(\chi_{6545}(212,\cdot)\) \(\chi_{6545}(247,\cdot)\) \(\chi_{6545}(478,\cdot)\) \(\chi_{6545}(807,\cdot)\) \(\chi_{6545}(933,\cdot)\) \(\chi_{6545}(977,\cdot)\) \(\chi_{6545}(1103,\cdot)\) \(\chi_{6545}(1402,\cdot)\) \(\chi_{6545}(1698,\cdot)\) \(\chi_{6545}(2293,\cdot)\) \(\chi_{6545}(2627,\cdot)\) \(\chi_{6545}(2797,\cdot)\) \(\chi_{6545}(2858,\cdot)\) \(\chi_{6545}(3028,\cdot)\) \(\chi_{6545}(3392,\cdot)\) \(\chi_{6545}(3623,\cdot)\) \(\chi_{6545}(3782,\cdot)\) \(\chi_{6545}(3908,\cdot)\) \(\chi_{6545}(3952,\cdot)\) \(\chi_{6545}(3987,\cdot)\) \(\chi_{6545}(4218,\cdot)\) \(\chi_{6545}(4503,\cdot)\) \(\chi_{6545}(4547,\cdot)\) \(\chi_{6545}(4673,\cdot)\) \(\chi_{6545}(5098,\cdot)\) \(\chi_{6545}(5142,\cdot)\) \(\chi_{6545}(5602,\cdot)\) \(\chi_{6545}(5833,\cdot)\) \(\chi_{6545}(6197,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((5237,3741,596,4236)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(18\)
\( \chi_{ 6545 }(1698, a) \) \(-1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{107}{120}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{19}{60}\right)\)
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)
 
\( \chi_{ 6545 }(1698,a) \;\) at \(\;a = \) e.g. 2