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Properties

Label 6555.2828
Modulus $6555$
Conductor $6555$
Order $36$
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(6555, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([18,27,8,18]))
 
Copy content gp:[g,chi] = znchar(Mod(2828, 6555))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6555.2828");
 

Basic properties

Modulus: \(6555\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6555\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
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 6555.df

\(\chi_{6555}(137,\cdot)\) \(\chi_{6555}(758,\cdot)\) \(\chi_{6555}(1448,\cdot)\) \(\chi_{6555}(1517,\cdot)\) \(\chi_{6555}(2552,\cdot)\) \(\chi_{6555}(2828,\cdot)\) \(\chi_{6555}(2897,\cdot)\) \(\chi_{6555}(3863,\cdot)\) \(\chi_{6555}(4208,\cdot)\) \(\chi_{6555}(4622,\cdot)\) \(\chi_{6555}(5933,\cdot)\) \(\chi_{6555}(6002,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((2186,1312,4486,856)\) → \((-1,-i,e\left(\frac{2}{9}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 6555 }(2828, a) \) \(-1\)\(1\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{5}{36}\right)\)
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)
 
\( \chi_{ 6555 }(2828,a) \;\) at \(\;a = \) e.g. 2