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Properties

Label 8400.6721
Modulus $8400$
Conductor $25$
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(8400, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,0,6,0]))
 
Copy content pari:[g,chi] = znchar(Mod(6721,8400))
 

Basic properties

Modulus: \(8400\)
Conductor: \(25\)
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 \(\chi_{25}(21,\cdot)\)
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 8400.dd

\(\chi_{8400}(1681,\cdot)\) \(\chi_{8400}(3361,\cdot)\) \(\chi_{8400}(5041,\cdot)\) \(\chi_{8400}(6721,\cdot)\)

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: 5.5.390625.1

Values on generators

\((3151,2101,2801,5377,3601)\) → \((1,1,1,e\left(\frac{3}{5}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8400 }(6721, a) \) \(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8400 }(6721,a) \;\) at \(\;a = \) e.g. 2