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Properties

Label 4256.1927
Modulus $4256$
Conductor $2128$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4256, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,3,4,2]))
 
Copy content pari:[g,chi] = znchar(Mod(1927,4256))
 

Basic properties

Modulus: \(4256\)
Conductor: \(2128\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2128}(331,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 4256.dy

\(\chi_{4256}(487,\cdot)\) \(\chi_{4256}(1927,\cdot)\) \(\chi_{4256}(2615,\cdot)\) \(\chi_{4256}(4055,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: Number field defined by a degree 12 polynomial

Values on generators

\((799,2661,3041,3137)\) → \((-1,i,e\left(\frac{1}{3}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 4256 }(1927, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4256 }(1927,a) \;\) at \(\;a = \) e.g. 2