Properties

Conductor 29
Order 28
Real No
Primitive Yes
Parity Odd
Orbit Label 29.f

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(29)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,29))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 29
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 28
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 = 29.f
Orbit index = 6

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 ]

\(\chi_{29}(2,\cdot)\) \(\chi_{29}(3,\cdot)\) \(\chi_{29}(8,\cdot)\) \(\chi_{29}(10,\cdot)\) \(\chi_{29}(11,\cdot)\) \(\chi_{29}(14,\cdot)\) \(\chi_{29}(15,\cdot)\) \(\chi_{29}(18,\cdot)\) \(\chi_{29}(19,\cdot)\) \(\chi_{29}(21,\cdot)\) \(\chi_{29}(26,\cdot)\) \(\chi_{29}(27,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{5}{28}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{13}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{28})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 29 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{29}(3,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) e\left(\frac{2r}{29}\right) = 1.8410703114+5.0606778309i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 29 }(3,·),\chi_{ 29 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{29}(3,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) \chi_{29}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 29 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{29}(3,·)) = \sum_{r \in \Z/29\Z} \chi_{29}(3,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right) = 0.4484787206+-0.713749795i \)