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Properties

Label 20.17
Modulus $20$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(17,20))
 

Basic properties

Modulus: \(20\)
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: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 20.f

\(\chi_{20}(13,\cdot)\) \(\chi_{20}(17,\cdot)\)

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

\((11,17)\) → \((1,i)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)
\( \chi_{ 20 }(17, a) \) \(-1\)\(1\)\(-i\)\(i\)\(-1\)\(1\)\(-i\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 20 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 20 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 20 }(17,·),\chi_{ 20 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 20 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2