Properties

Modulus 64
Conductor 64
Order 16
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 64.i

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(64)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(29,64))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 64
Conductor = 64
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 16
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 64.i
Orbit index = 9

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{64}(5,\cdot)\) \(\chi_{64}(13,\cdot)\) \(\chi_{64}(21,\cdot)\) \(\chi_{64}(29,\cdot)\) \(\chi_{64}(37,\cdot)\) \(\chi_{64}(45,\cdot)\) \(\chi_{64}(53,\cdot)\) \(\chi_{64}(61,\cdot)\)

Values on generators

\((63,5)\) → \((1,e\left(\frac{11}{16}\right))\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(-i\)\(i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{16}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{16})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 64 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{64}(29,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(29,r) e\left(\frac{r}{32}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 64 }(29,·),\chi_{ 64 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{64}(29,\cdot),\chi_{64}(1,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(29,r) \chi_{64}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 64 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{64}(29,·)) = \sum_{r \in \Z/64\Z} \chi_{64}(29,r) e\left(\frac{1 r + 2 r^{-1}}{64}\right) = 7.6555226859+2.322277418i \)