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Properties

Label 725.568
Modulus $725$
Conductor $145$
Order $4$
Real no
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(725, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([3,3]))
 
Copy content pari:[g,chi] = znchar(Mod(568,725))
 

Basic properties

Modulus: \(725\)
Conductor: \(145\)
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 \(\chi_{145}(133,\cdot)\)
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 725.e

\(\chi_{725}(157,\cdot)\) \(\chi_{725}(568,\cdot)\)

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: 4.4.3048625.2

Values on generators

\((552,176)\) → \((-i,-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 725 }(568, a) \) \(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-i\)\(-1\)\(1\)\(-i\)\(1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 725 }(568,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 725 }(568,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 725 }(568,·),\chi_{ 725 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 725 }(568,·)) \;\) at \(\; a,b = \) e.g. 1,2