Properties

Conductor 179
Order 89
Real No
Primitive Yes
Parity Even
Orbit Label 179.c

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(179)
sage: chi = H[22]
pari: [g,chi] = znchar(Mod(22,179))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 179
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 89
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 = Even
Orbit label = 179.c
Orbit index = 3

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 ]

\(\chi_{179}(3,\cdot)\) \(\chi_{179}(4,\cdot)\) \(\chi_{179}(5,\cdot)\) \(\chi_{179}(9,\cdot)\) \(\chi_{179}(12,\cdot)\) \(\chi_{179}(13,\cdot)\) \(\chi_{179}(14,\cdot)\) \(\chi_{179}(15,\cdot)\) \(\chi_{179}(16,\cdot)\) \(\chi_{179}(17,\cdot)\) \(\chi_{179}(19,\cdot)\) \(\chi_{179}(20,\cdot)\) \(\chi_{179}(22,\cdot)\) \(\chi_{179}(25,\cdot)\) \(\chi_{179}(27,\cdot)\) \(\chi_{179}(29,\cdot)\) \(\chi_{179}(31,\cdot)\) \(\chi_{179}(36,\cdot)\) \(\chi_{179}(39,\cdot)\) \(\chi_{179}(42,\cdot)\) \(\chi_{179}(43,\cdot)\) \(\chi_{179}(45,\cdot)\) \(\chi_{179}(46,\cdot)\) \(\chi_{179}(47,\cdot)\) \(\chi_{179}(48,\cdot)\) \(\chi_{179}(49,\cdot)\) \(\chi_{179}(51,\cdot)\) \(\chi_{179}(52,\cdot)\) \(\chi_{179}(56,\cdot)\) \(\chi_{179}(57,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{8}{89}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{8}{89}\right)\)\(e\left(\frac{63}{89}\right)\)\(e\left(\frac{16}{89}\right)\)\(e\left(\frac{36}{89}\right)\)\(e\left(\frac{71}{89}\right)\)\(e\left(\frac{33}{89}\right)\)\(e\left(\frac{24}{89}\right)\)\(e\left(\frac{37}{89}\right)\)\(e\left(\frac{44}{89}\right)\)\(e\left(\frac{31}{89}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{89})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 179 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{179}(22,\cdot)) = \sum_{r\in \Z/179\Z} \chi_{179}(22,r) e\left(\frac{2r}{179}\right) = 2.6675811167+13.1104542631i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 179 }(22,·),\chi_{ 179 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{179}(22,\cdot),\chi_{179}(1,\cdot)) = \sum_{r\in \Z/179\Z} \chi_{179}(22,r) \chi_{179}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 179 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{179}(22,·)) = \sum_{r \in \Z/179\Z} \chi_{179}(22,r) e\left(\frac{1 r + 2 r^{-1}}{179}\right) = 12.6597736651+3.6731588107i \)