Properties

Label 287.263
Modulus $287$
Conductor $287$
Order $120$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(287, base_ring=CyclotomicField(120))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([80,99]))
 
pari: [g,chi] = znchar(Mod(263,287))
 

Basic properties

Modulus: \(287\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 287.bf

\(\chi_{287}(11,\cdot)\) \(\chi_{287}(30,\cdot)\) \(\chi_{287}(53,\cdot)\) \(\chi_{287}(58,\cdot)\) \(\chi_{287}(60,\cdot)\) \(\chi_{287}(65,\cdot)\) \(\chi_{287}(67,\cdot)\) \(\chi_{287}(88,\cdot)\) \(\chi_{287}(93,\cdot)\) \(\chi_{287}(95,\cdot)\) \(\chi_{287}(116,\cdot)\) \(\chi_{287}(130,\cdot)\) \(\chi_{287}(135,\cdot)\) \(\chi_{287}(142,\cdot)\) \(\chi_{287}(149,\cdot)\) \(\chi_{287}(151,\cdot)\) \(\chi_{287}(158,\cdot)\) \(\chi_{287}(170,\cdot)\) \(\chi_{287}(177,\cdot)\) \(\chi_{287}(179,\cdot)\) \(\chi_{287}(186,\cdot)\) \(\chi_{287}(193,\cdot)\) \(\chi_{287}(198,\cdot)\) \(\chi_{287}(212,\cdot)\) \(\chi_{287}(233,\cdot)\) \(\chi_{287}(235,\cdot)\) \(\chi_{287}(240,\cdot)\) \(\chi_{287}(261,\cdot)\) \(\chi_{287}(263,\cdot)\) \(\chi_{287}(268,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{33}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{17}{120}\right)\)\(e\left(\frac{73}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 287 }(263,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(263,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 287 }(263,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(263,·)) \;\) at \(\; a,b = \) e.g. 1,2