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Properties

Label 2576.261
Modulus $2576$
Conductor $2576$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

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(2576, base_ring=CyclotomicField(132)) M = H._module chi = DirichletCharacter(H, M([0,33,44,36]))
 
Copy content gp:[g,chi] = znchar(Mod(261, 2576))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2576.261");
 

Basic properties

Modulus: \(2576\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2576\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(132\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2576.dm

\(\chi_{2576}(165,\cdot)\) \(\chi_{2576}(261,\cdot)\) \(\chi_{2576}(317,\cdot)\) \(\chi_{2576}(445,\cdot)\) \(\chi_{2576}(485,\cdot)\) \(\chi_{2576}(501,\cdot)\) \(\chi_{2576}(541,\cdot)\) \(\chi_{2576}(653,\cdot)\) \(\chi_{2576}(669,\cdot)\) \(\chi_{2576}(725,\cdot)\) \(\chi_{2576}(765,\cdot)\) \(\chi_{2576}(821,\cdot)\) \(\chi_{2576}(837,\cdot)\) \(\chi_{2576}(877,\cdot)\) \(\chi_{2576}(933,\cdot)\) \(\chi_{2576}(949,\cdot)\) \(\chi_{2576}(1005,\cdot)\) \(\chi_{2576}(1061,\cdot)\) \(\chi_{2576}(1117,\cdot)\) \(\chi_{2576}(1269,\cdot)\) \(\chi_{2576}(1453,\cdot)\) \(\chi_{2576}(1549,\cdot)\) \(\chi_{2576}(1605,\cdot)\) \(\chi_{2576}(1733,\cdot)\) \(\chi_{2576}(1773,\cdot)\) \(\chi_{2576}(1789,\cdot)\) \(\chi_{2576}(1829,\cdot)\) \(\chi_{2576}(1941,\cdot)\) \(\chi_{2576}(1957,\cdot)\) \(\chi_{2576}(2013,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((2255,645,1473,1569)\) → \((1,i,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(25\)\(27\)
\( \chi_{ 2576 }(261, a) \) \(1\)\(1\)\(e\left(\frac{59}{132}\right)\)\(e\left(\frac{25}{132}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{5}{132}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{67}{132}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{15}{44}\right)\)
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)
 
\( \chi_{ 2576 }(261,a) \;\) at \(\;a = \) e.g. 2