Properties

Label 529.27
Modulus $529$
Conductor $529$
Order $253$
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(529, base_ring=CyclotomicField(506))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([48]))
 
pari: [g,chi] = znchar(Mod(27,529))
 

Basic properties

Modulus: \(529\)
Conductor: \(529\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(253\)
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 529.g

\(\chi_{529}(2,\cdot)\) \(\chi_{529}(3,\cdot)\) \(\chi_{529}(4,\cdot)\) \(\chi_{529}(6,\cdot)\) \(\chi_{529}(8,\cdot)\) \(\chi_{529}(9,\cdot)\) \(\chi_{529}(12,\cdot)\) \(\chi_{529}(13,\cdot)\) \(\chi_{529}(16,\cdot)\) \(\chi_{529}(18,\cdot)\) \(\chi_{529}(25,\cdot)\) \(\chi_{529}(26,\cdot)\) \(\chi_{529}(27,\cdot)\) \(\chi_{529}(29,\cdot)\) \(\chi_{529}(31,\cdot)\) \(\chi_{529}(32,\cdot)\) \(\chi_{529}(35,\cdot)\) \(\chi_{529}(36,\cdot)\) \(\chi_{529}(39,\cdot)\) \(\chi_{529}(41,\cdot)\) \(\chi_{529}(48,\cdot)\) \(\chi_{529}(49,\cdot)\) \(\chi_{529}(50,\cdot)\) \(\chi_{529}(52,\cdot)\) \(\chi_{529}(54,\cdot)\) \(\chi_{529}(55,\cdot)\) \(\chi_{529}(58,\cdot)\) \(\chi_{529}(59,\cdot)\) \(\chi_{529}(62,\cdot)\) \(\chi_{529}(64,\cdot)\) ...

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: $\Q(\zeta_{253})$
Fixed field: Number field defined by a degree 253 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{24}{253}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{246}{253}\right)\)\(e\left(\frac{131}{253}\right)\)\(e\left(\frac{239}{253}\right)\)\(e\left(\frac{24}{253}\right)\)\(e\left(\frac{124}{253}\right)\)\(e\left(\frac{60}{253}\right)\)\(e\left(\frac{232}{253}\right)\)\(e\left(\frac{9}{253}\right)\)\(e\left(\frac{17}{253}\right)\)\(e\left(\frac{150}{253}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 529 }(27,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 529 }(27,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 529 }(27,·),\chi_{ 529 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 529 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2