↑

Properties

Label 961.388
Modulus $961$
Conductor $31$
Order $5$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([2]))
 
Copy content pari:[g,chi] = znchar(Mod(388,961))
 

Basic properties

Modulus: \(961\)
Conductor: \(31\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{31}(16,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 961.d

\(\chi_{961}(374,\cdot)\) \(\chi_{961}(388,\cdot)\) \(\chi_{961}(531,\cdot)\) \(\chi_{961}(628,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 5.5.923521.1

Values on generators

\(3\) → \(e\left(\frac{1}{5}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 961 }(388, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 961 }(388,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 961 }(388,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 961 }(388,·),\chi_{ 961 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 961 }(388,·)) \;\) at \(\; a,b = \) e.g. 1,2