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Properties

Label 4998.4115
Modulus $4998$
Conductor $21$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4998, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(4115,4998))
 

Basic properties

Modulus: \(4998\)
Conductor: \(21\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{21}(20,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 4998.f

\(\chi_{4998}(4115,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{21}) \)

Values on generators

\((1667,2551,4117)\) → \((-1,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4998 }(4115, a) \) \(1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4998 }(4115,a) \;\) at \(\;a = \) e.g. 2