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Properties

Label 1050.1007
Modulus $1050$
Conductor $105$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,1,2]))
 
Copy content pari:[g,chi] = znchar(Mod(1007,1050))
 

Basic properties

Modulus: \(1050\)
Conductor: \(105\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{105}(62,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1050.k

\(\chi_{1050}(293,\cdot)\) \(\chi_{1050}(1007,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: 4.0.55125.1

Values on generators

\((701,127,451)\) → \((-1,i,-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1050 }(1007, a) \) \(-1\)\(1\)\(-1\)\(i\)\(i\)\(1\)\(i\)\(1\)\(-1\)\(i\)\(1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1050 }(1007,a) \;\) at \(\;a = \) e.g. 2