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Properties

Label 2057.g
Modulus $2057$
Conductor $11$
Order $5$
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(2057, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([4,0])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(511,2057)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2057\)
Conductor: \(11\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 11.c
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)
 

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{11})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{2057}(511,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(1\) \(1\)
\(\chi_{2057}(1412,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(1\)
\(\chi_{2057}(1582,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(1\)
\(\chi_{2057}(1939,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(1\)