Properties

Label 483.85
Modulus $483$
Conductor $23$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

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([0,0,8]))
 
pari: [g,chi] = znchar(Mod(85,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 483.q

\(\chi_{483}(64,\cdot)\) \(\chi_{483}(85,\cdot)\) \(\chi_{483}(127,\cdot)\) \(\chi_{483}(169,\cdot)\) \(\chi_{483}(190,\cdot)\) \(\chi_{483}(211,\cdot)\) \(\chi_{483}(232,\cdot)\) \(\chi_{483}(358,\cdot)\) \(\chi_{483}(400,\cdot)\) \(\chi_{483}(463,\cdot)\)

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: \(\Q(\zeta_{23})^+\)

Values on generators

\((323,346,442)\) → \((1,1,e\left(\frac{4}{11}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(85,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{483}(85,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(85,r) e\left(\frac{2r}{483}\right) = -0.0564765018+-4.7954989735i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(85,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{483}(85,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(85,r) \chi_{483}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{483}(85,·)) = \sum_{r \in \Z/483\Z} \chi_{483}(85,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right) = -7.4170842393+8.5597715151i \)