Properties

Conductor 229
Order 76
Real No
Primitive Yes
Parity Odd
Orbit Label 229.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(229)
sage: chi = H[141]
pari: [g,chi] = znchar(Mod(141,229))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 229
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 76
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 = 229.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 ]

\(\chi_{229}(2,\cdot)\) \(\chi_{229}(8,\cdot)\) \(\chi_{229}(13,\cdot)\) \(\chi_{229}(21,\cdot)\) \(\chi_{229}(22,\cdot)\) \(\chi_{229}(30,\cdot)\) \(\chi_{229}(32,\cdot)\) \(\chi_{229}(34,\cdot)\) \(\chi_{229}(52,\cdot)\) \(\chi_{229}(54,\cdot)\) \(\chi_{229}(84,\cdot)\) \(\chi_{229}(86,\cdot)\) \(\chi_{229}(88,\cdot)\) \(\chi_{229}(93,\cdot)\) \(\chi_{229}(101,\cdot)\) \(\chi_{229}(106,\cdot)\) \(\chi_{229}(109,\cdot)\) \(\chi_{229}(114,\cdot)\) \(\chi_{229}(115,\cdot)\) \(\chi_{229}(120,\cdot)\) \(\chi_{229}(123,\cdot)\) \(\chi_{229}(128,\cdot)\) \(\chi_{229}(136,\cdot)\) \(\chi_{229}(141,\cdot)\) \(\chi_{229}(143,\cdot)\) \(\chi_{229}(145,\cdot)\) \(\chi_{229}(175,\cdot)\) \(\chi_{229}(177,\cdot)\) \(\chi_{229}(195,\cdot)\) \(\chi_{229}(197,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{37}{76}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{17}{76}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{17}{38}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{37}{76}\right)\)\(e\left(\frac{7}{76}\right)\)\(e\left(\frac{51}{76}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{71}{76}\right)\)\(e\left(\frac{33}{38}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{76})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 229 }(141,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{229}(141,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(141,r) e\left(\frac{2r}{229}\right) = -4.5073315215+14.445897776i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 229 }(141,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{229}(141,\cdot),\chi_{229}(1,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(141,r) \chi_{229}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 229 }(141,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{229}(141,·)) = \sum_{r \in \Z/229\Z} \chi_{229}(141,r) e\left(\frac{1 r + 2 r^{-1}}{229}\right) = -19.2541151085+22.7332907512i \)