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Properties

Label 720.577
Modulus $720$
Conductor $5$
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(720, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(577,720))
 

Basic properties

Modulus: \(720\)
Conductor: \(5\)
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 \(\chi_{5}(2,\cdot)\)
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 720.bh

\(\chi_{720}(433,\cdot)\) \(\chi_{720}(577,\cdot)\)

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

Values on generators

\((271,181,641,577)\) → \((1,1,1,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 720 }(577, a) \) \(-1\)\(1\)\(i\)\(1\)\(-i\)\(i\)\(-1\)\(-i\)\(-1\)\(1\)\(i\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 720 }(577,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 720 }(577,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 720 }(577,·),\chi_{ 720 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 720 }(577,·)) \;\) at \(\; a,b = \) e.g. 1,2