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Properties

Label 1712.641
Modulus $1712$
Conductor $107$
Order $2$
Real yes
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1712, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(641,1712))
 

Basic properties

Modulus: \(1712\)
Conductor: \(107\)
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 \(\chi_{107}(106,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1712.g

\(\chi_{1712}(641,\cdot)\)

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: \(\Q(\sqrt{-107}) \)

Values on generators

\((1071,1285,1393)\) → \((1,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1712 }(641, a) \) \(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1712 }(641,a) \;\) at \(\;a = \) e.g. 2