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Properties

Label 6800.1121
Modulus $6800$
Conductor $425$
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(6800, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,6,5]))
 
Copy content pari:[g,chi] = znchar(Mod(1121,6800))
 

Basic properties

Modulus: \(6800\)
Conductor: \(425\)
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 \(\chi_{425}(271,\cdot)\)
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 6800.ek

\(\chi_{6800}(1121,\cdot)\) \(\chi_{6800}(2481,\cdot)\) \(\chi_{6800}(3841,\cdot)\) \(\chi_{6800}(6561,\cdot)\)

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: 10.10.216652984619140625.1

Values on generators

\((5951,1701,2177,1601)\) → \((1,1,e\left(\frac{3}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6800 }(1121, a) \) \(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(-1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{7}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 6800 }(1121,a) \;\) at \(\;a = \) e.g. 2