Properties

Label 483.4
Modulus $483$
Conductor $161$
Order $33$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: 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([0,44,12]))
 
pari: [g,chi] = znchar(Mod(4,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 483.y

\(\chi_{483}(4,\cdot)\) \(\chi_{483}(16,\cdot)\) \(\chi_{483}(25,\cdot)\) \(\chi_{483}(58,\cdot)\) \(\chi_{483}(100,\cdot)\) \(\chi_{483}(121,\cdot)\) \(\chi_{483}(142,\cdot)\) \(\chi_{483}(151,\cdot)\) \(\chi_{483}(163,\cdot)\) \(\chi_{483}(193,\cdot)\) \(\chi_{483}(256,\cdot)\) \(\chi_{483}(289,\cdot)\) \(\chi_{483}(331,\cdot)\) \(\chi_{483}(340,\cdot)\) \(\chi_{483}(361,\cdot)\) \(\chi_{483}(394,\cdot)\) \(\chi_{483}(403,\cdot)\) \(\chi_{483}(445,\cdot)\) \(\chi_{483}(466,\cdot)\) \(\chi_{483}(478,\cdot)\)

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: 33.33.277966181338944111003326058293667039541136678070715028736001.1

Values on generators

\((323,346,442)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{2}{33}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{483}(4,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(4,r) e\left(\frac{2r}{483}\right) = -5.8904074034+11.2384652254i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(4,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{483}(4,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(4,r) \chi_{483}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{483}(4,·)) = \sum_{r \in \Z/483\Z} \chi_{483}(4,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right) = 7.3496719946+-10.3211701533i \)