Properties

Conductor 23
Order 22
Real No
Primitive Yes
Parity Odd
Orbit Label 23.d

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(23)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,23))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 23
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 22
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 = 23.d
Orbit index = 4

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 ]

\(\chi_{23}(5,\cdot)\) \(\chi_{23}(7,\cdot)\) \(\chi_{23}(10,\cdot)\) \(\chi_{23}(11,\cdot)\) \(\chi_{23}(14,\cdot)\) \(\chi_{23}(15,\cdot)\) \(\chi_{23}(17,\cdot)\) \(\chi_{23}(19,\cdot)\) \(\chi_{23}(20,\cdot)\) \(\chi_{23}(21,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{9}{22}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{15}{22}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{11})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 23 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{23}(11,\cdot)) = \sum_{r\in \Z/23\Z} \chi_{23}(11,r) e\left(\frac{2r}{23}\right) = 4.4835611503+-1.7022571519i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 23 }(11,·),\chi_{ 23 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{23}(11,\cdot),\chi_{23}(1,\cdot)) = \sum_{r\in \Z/23\Z} \chi_{23}(11,r) \chi_{23}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 23 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{23}(11,·)) = \sum_{r \in \Z/23\Z} \chi_{23}(11,r) e\left(\frac{1 r + 2 r^{-1}}{23}\right) = 3.5133078303+5.4668136935i \)