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Properties

Label 1728.593
Modulus $1728$
Conductor $48$
Order $4$
Real no
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(1728, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(593,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(48\)
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 \(\chi_{48}(29,\cdot)\)
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 1728.j

\(\chi_{1728}(593,\cdot)\) \(\chi_{1728}(1457,\cdot)\)

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: 4.0.18432.2

Values on generators

\((703,325,1217)\) → \((1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1728 }(593, a) \) \(-1\)\(1\)\(i\)\(-1\)\(i\)\(i\)\(-1\)\(i\)\(1\)\(-1\)\(-i\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(593,a) \;\) at \(\;a = \) e.g. 2