Properties

Conductor 13
Order 6
Real No
Primitive No
Parity Even
Orbit Label 39.j

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(39)
sage: chi = H[4]
pari: [g,chi] = znchar(Mod(4,39))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 13
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 6
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 = 39.j
Orbit index = 10

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 ]

\(\chi_{39}(4,\cdot)\) \(\chi_{39}(10,\cdot)\)

Inducing primitive character

\(\chi_{13}(4,\cdot)\)

Values on generators

\((14,28)\) → \((1,e\left(\frac{1}{6}\right))\)

Values

-11245781011141617
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 39 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{39}(4,\cdot)) = \sum_{r\in \Z/39\Z} \chi_{39}(4,r) e\left(\frac{2r}{39}\right) = 3.0991246837+1.842668227i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 39 }(4,·),\chi_{ 39 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{39}(4,\cdot),\chi_{39}(1,\cdot)) = \sum_{r\in \Z/39\Z} \chi_{39}(4,r) \chi_{39}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 39 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{39}(4,·)) = \sum_{r \in \Z/39\Z} \chi_{39}(4,r) e\left(\frac{1 r + 2 r^{-1}}{39}\right) = 11.1383870585+6.4307507665i \)