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Properties

Label 1256.261
Modulus $1256$
Conductor $1256$
Order $156$
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(1256, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([0,78,137]))
 
Copy content gp:[g,chi] = znchar(Mod(261, 1256))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1256.261");
 

Basic properties

Modulus: \(1256\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1256\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 1256.bu

\(\chi_{1256}(5,\cdot)\) \(\chi_{1256}(21,\cdot)\) \(\chi_{1256}(53,\cdot)\) \(\chi_{1256}(61,\cdot)\) \(\chi_{1256}(69,\cdot)\) \(\chi_{1256}(77,\cdot)\) \(\chi_{1256}(85,\cdot)\) \(\chi_{1256}(133,\cdot)\) \(\chi_{1256}(181,\cdot)\) \(\chi_{1256}(229,\cdot)\) \(\chi_{1256}(237,\cdot)\) \(\chi_{1256}(245,\cdot)\) \(\chi_{1256}(253,\cdot)\) \(\chi_{1256}(261,\cdot)\) \(\chi_{1256}(293,\cdot)\) \(\chi_{1256}(309,\cdot)\) \(\chi_{1256}(357,\cdot)\) \(\chi_{1256}(397,\cdot)\) \(\chi_{1256}(405,\cdot)\) \(\chi_{1256}(437,\cdot)\) \(\chi_{1256}(445,\cdot)\) \(\chi_{1256}(453,\cdot)\) \(\chi_{1256}(477,\cdot)\) \(\chi_{1256}(509,\cdot)\) \(\chi_{1256}(533,\cdot)\) \(\chi_{1256}(541,\cdot)\) \(\chi_{1256}(565,\cdot)\) \(\chi_{1256}(573,\cdot)\) \(\chi_{1256}(613,\cdot)\) \(\chi_{1256}(701,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((943,629,633)\) → \((1,-1,e\left(\frac{137}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1256 }(261, a) \) \(-1\)\(1\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{59}{156}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{139}{156}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{95}{156}\right)\)
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)
 
\( \chi_{ 1256 }(261,a) \;\) at \(\;a = \) e.g. 2