Properties

Conductor 63
Order 6
Real No
Primitive Yes
Parity Odd
Orbit Label 63.n

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(63)
sage: chi = H[32]
pari: [g,chi] = znchar(Mod(32,63))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 63
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 = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 63.n
Orbit index = 14

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 ]

\(\chi_{63}(2,\cdot)\) \(\chi_{63}(32,\cdot)\)

Values on generators

\((29,10)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right))\)

Values

-112458101113161719
\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)
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)
\( \tau_{ a }( \chi_{ 63 }(32,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{63}(32,\cdot)) = \sum_{r\in \Z/63\Z} \chi_{63}(32,r) e\left(\frac{2r}{63}\right) = -3.1816234537+-7.2716760241i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 63 }(32,·),\chi_{ 63 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{63}(32,\cdot),\chi_{63}(1,\cdot)) = \sum_{r\in \Z/63\Z} \chi_{63}(32,r) \chi_{63}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 63 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{63}(32,·)) = \sum_{r \in \Z/63\Z} \chi_{63}(32,r) e\left(\frac{1 r + 2 r^{-1}}{63}\right) = -2.0353438018+3.5253188757i \)