Properties

Label 483.199
Modulus $483$
Conductor $161$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,51]))
 
pari: [g,chi] = znchar(Mod(199,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 483.bf

\(\chi_{483}(10,\cdot)\) \(\chi_{483}(19,\cdot)\) \(\chi_{483}(40,\cdot)\) \(\chi_{483}(61,\cdot)\) \(\chi_{483}(103,\cdot)\) \(\chi_{483}(136,\cdot)\) \(\chi_{483}(145,\cdot)\) \(\chi_{483}(157,\cdot)\) \(\chi_{483}(166,\cdot)\) \(\chi_{483}(178,\cdot)\) \(\chi_{483}(199,\cdot)\) \(\chi_{483}(241,\cdot)\) \(\chi_{483}(250,\cdot)\) \(\chi_{483}(283,\cdot)\) \(\chi_{483}(304,\cdot)\) \(\chi_{483}(313,\cdot)\) \(\chi_{483}(355,\cdot)\) \(\chi_{483}(388,\cdot)\) \(\chi_{483}(451,\cdot)\) \(\chi_{483}(481,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((323,346,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{17}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{14}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 483 }(199,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(199,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(199,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(199,·)) \;\) at \(\; a,b = \) e.g. 1,2