Properties

Conductor 32
Order 8
Real No
Primitive No
Parity Even
Orbit Label 64.g

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(64)
sage: chi = H[25]
pari: [g,chi] = znchar(Mod(25,64))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 32
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 8
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 64.g
Orbit index = 7

Galois orbit

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

\(\chi_{64}(9,\cdot)\) \(\chi_{64}(25,\cdot)\) \(\chi_{64}(41,\cdot)\) \(\chi_{64}(57,\cdot)\)

Inducing primitive character

\(\chi_{32}(5,\cdot)\)

Values on generators

\((63,5)\) → \((1,e\left(\frac{1}{8}\right))\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 64 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{64}(25,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(25,r) e\left(\frac{r}{32}\right) = 6.2855596671+9.4070048194i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 64 }(25,·),\chi_{ 64 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{64}(25,\cdot),\chi_{64}(1,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(25,r) \chi_{64}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 64 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{64}(25,·)) = \sum_{r \in \Z/64\Z} \chi_{64}(25,r) e\left(\frac{1 r + 2 r^{-1}}{64}\right) = 0.0 \)