Properties

Conductor 91
Order 12
Real No
Primitive Yes
Parity Odd
Orbit Label 91.x

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(91)
sage: chi = H[32]
pari: [g,chi] = znchar(Mod(32,91))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 91
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 12
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 = 91.x
Orbit index = 24

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 ]

\(\chi_{91}(2,\cdot)\) \(\chi_{91}(32,\cdot)\) \(\chi_{91}(37,\cdot)\) \(\chi_{91}(46,\cdot)\)

Values on generators

\((66,15)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{12}\right))\)

Values

-112345689101112
\(-1\)\(1\)\(-i\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 91 }(32,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{91}(32,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(32,r) e\left(\frac{2r}{91}\right) = 7.7328133792+-5.586018013i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 91 }(32,·),\chi_{ 91 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{91}(32,\cdot),\chi_{91}(1,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(32,r) \chi_{91}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 91 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{91}(32,·)) = \sum_{r \in \Z/91\Z} \chi_{91}(32,r) e\left(\frac{1 r + 2 r^{-1}}{91}\right) = -2.0333454457+-2.0333454457i \)