Properties

Label 529.231
Modulus $529$
Conductor $529$
Order $23$
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(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(231,529))
 

Basic properties

Modulus: \(529\)
Conductor: \(529\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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.e

\(\chi_{529}(24,\cdot)\) \(\chi_{529}(47,\cdot)\) \(\chi_{529}(70,\cdot)\) \(\chi_{529}(93,\cdot)\) \(\chi_{529}(116,\cdot)\) \(\chi_{529}(139,\cdot)\) \(\chi_{529}(162,\cdot)\) \(\chi_{529}(185,\cdot)\) \(\chi_{529}(208,\cdot)\) \(\chi_{529}(231,\cdot)\) \(\chi_{529}(254,\cdot)\) \(\chi_{529}(277,\cdot)\) \(\chi_{529}(300,\cdot)\) \(\chi_{529}(323,\cdot)\) \(\chi_{529}(346,\cdot)\) \(\chi_{529}(369,\cdot)\) \(\chi_{529}(392,\cdot)\) \(\chi_{529}(415,\cdot)\) \(\chi_{529}(438,\cdot)\) \(\chi_{529}(461,\cdot)\) \(\chi_{529}(484,\cdot)\) \(\chi_{529}(507,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: 23.23.824185149135487077883465900577270766354751717380230010246241.1

Values on generators

\(5\) → \(e\left(\frac{4}{23}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{22}{23}\right)\)\(e\left(\frac{2}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 529 }(231,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 529 }(231,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 529 }(231,·),\chi_{ 529 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 529 }(231,·)) \;\) at \(\; a,b = \) e.g. 1,2