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Properties

Label 5200.4081
Modulus $5200$
Conductor $325$
Order $10$
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(5200, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,4,5]))
 
Copy content pari:[g,chi] = znchar(Mod(4081,5200))
 

Basic properties

Modulus: \(5200\)
Conductor: \(325\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(10\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{325}(181,\cdot)\)
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 5200.dn

\(\chi_{5200}(961,\cdot)\) \(\chi_{5200}(3041,\cdot)\) \(\chi_{5200}(4081,\cdot)\) \(\chi_{5200}(5121,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.56654815673828125.1

Values on generators

\((1951,1301,4577,1601)\) → \((1,1,e\left(\frac{2}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 5200 }(4081, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5200 }(4081,a) \;\) at \(\;a = \) e.g. 2