Properties

Label 483.113
Modulus $483$
Conductor $69$
Order $22$
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,0,13]))
 
pari: [g,chi] = znchar(Mod(113,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(44,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 483.u

\(\chi_{483}(113,\cdot)\) \(\chi_{483}(134,\cdot)\) \(\chi_{483}(155,\cdot)\) \(\chi_{483}(176,\cdot)\) \(\chi_{483}(218,\cdot)\) \(\chi_{483}(260,\cdot)\) \(\chi_{483}(281,\cdot)\) \(\chi_{483}(365,\cdot)\) \(\chi_{483}(428,\cdot)\) \(\chi_{483}(470,\cdot)\)

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: \(\Q(\zeta_{69})^+\)

Values on generators

\((323,346,442)\) → \((-1,1,e\left(\frac{13}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{22}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(113,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{483}(113,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(113,r) e\left(\frac{2r}{483}\right) = 8.2887877736+-0.5440562872i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(113,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{483}(113,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(113,r) \chi_{483}(1,1-r) = 5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{483}(113,·)) = \sum_{r \in \Z/483\Z} \chi_{483}(113,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right) = 0.0 \)