Properties

Modulus 68
Conductor 17
Order 16
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 68.j

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(68)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5]))
 
pari: [g,chi] = znchar(Mod(5,68))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 68
Conductor = 17
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 = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 68.j
Orbit index = 10

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 ]
 

\(\chi_{68}(5,\cdot)\) \(\chi_{68}(29,\cdot)\) \(\chi_{68}(37,\cdot)\) \(\chi_{68}(41,\cdot)\) \(\chi_{68}(45,\cdot)\) \(\chi_{68}(57,\cdot)\) \(\chi_{68}(61,\cdot)\) \(\chi_{68}(65,\cdot)\)

Values on generators

\((35,37)\) → \((1,e\left(\frac{5}{16}\right))\)

Values

-113579111315192123
\(-1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{11}{16}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 68 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{68}(5,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(5,r) e\left(\frac{r}{34}\right) = -4.3609541362+-6.9987198131i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 68 }(5,·),\chi_{ 68 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{68}(5,\cdot),\chi_{68}(1,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(5,r) \chi_{68}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 68 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{68}(5,·)) = \sum_{r \in \Z/68\Z} \chi_{68}(5,r) e\left(\frac{1 r + 2 r^{-1}}{68}\right) = -0.0 \)