Properties

Label 483.10
Modulus $483$
Conductor $161$
Order $66$
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([0,11,9]))
 
pari: [g,chi] = znchar(Mod(10,483))
 

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 \(\chi_{161}(10,\cdot)\)
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

\((323,346,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{29}{33}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{483}(10,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(10,r) e\left(\frac{2r}{483}\right) = 0.4902815379+12.6791018615i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(10,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{483}(10,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(10,r) \chi_{483}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{483}(10,·)) = \sum_{r \in \Z/483\Z} \chi_{483}(10,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right) = 2.4203017533+-6.9930026627i \)