Properties

Label 287.221
Modulus $287$
Conductor $287$
Order $15$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(287, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,18]))
 
pari: [g,chi] = znchar(Mod(221,287))
 

Basic properties

Modulus: \(287\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 287.s

\(\chi_{287}(16,\cdot)\) \(\chi_{287}(18,\cdot)\) \(\chi_{287}(37,\cdot)\) \(\chi_{287}(51,\cdot)\) \(\chi_{287}(100,\cdot)\) \(\chi_{287}(221,\cdot)\) \(\chi_{287}(242,\cdot)\) \(\chi_{287}(256,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 15.15.6373627540905023204410809169.1

Values on generators

\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 287 }(221,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(221,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 287 }(221,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(221,·)) \;\) at \(\; a,b = \) e.g. 1,2