Properties

Label 483.41
Modulus $483$
Conductor $483$
Order $22$
Real no
Primitive yes
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,12]))
 
pari: [g,chi] = znchar(Mod(41,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 483.v

\(\chi_{483}(41,\cdot)\) \(\chi_{483}(62,\cdot)\) \(\chi_{483}(104,\cdot)\) \(\chi_{483}(146,\cdot)\) \(\chi_{483}(167,\cdot)\) \(\chi_{483}(188,\cdot)\) \(\chi_{483}(209,\cdot)\) \(\chi_{483}(335,\cdot)\) \(\chi_{483}(377,\cdot)\) \(\chi_{483}(440,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.22.601130775140836298755595442714814879781421.1

Values on generators

\((323,346,442)\) → \((-1,-1,e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{483}(41,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(41,r) e\left(\frac{2r}{483}\right) = 21.6756233438+3.628684701i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(41,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{483}(41,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(41,r) \chi_{483}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{483}(41,·)) = \sum_{r \in \Z/483\Z} \chi_{483}(41,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right) = 0.0 \)