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Properties

Label 4400.4049
Modulus $4400$
Conductor $5$
Order $2$
Real yes
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(4400, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([0,0,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(4049,4400))
 

Basic properties

Modulus: \(4400\)
Conductor: \(5\)
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 \(\chi_{5}(4,\cdot)\)
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 4400.b

\(\chi_{4400}(4049,\cdot)\)

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: \(\Q(\sqrt{5}) \)

Values on generators

\((2751,3301,177,1201)\) → \((1,1,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4400 }(4049, a) \) \(1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4400 }(4049,a) \;\) at \(\;a = \) e.g. 2