Properties

Conductor 89
Order 88
Real No
Primitive Yes
Parity Odd
Orbit Label 89.h

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(89)
sage: chi = H[61]
pari: [g,chi] = znchar(Mod(61,89))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 89
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 88
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 = 89.h
Orbit index = 8

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 ]

\(\chi_{89}(3,\cdot)\) \(\chi_{89}(6,\cdot)\) \(\chi_{89}(7,\cdot)\) \(\chi_{89}(13,\cdot)\) \(\chi_{89}(14,\cdot)\) \(\chi_{89}(15,\cdot)\) \(\chi_{89}(19,\cdot)\) \(\chi_{89}(23,\cdot)\) \(\chi_{89}(24,\cdot)\) \(\chi_{89}(26,\cdot)\) \(\chi_{89}(27,\cdot)\) \(\chi_{89}(28,\cdot)\) \(\chi_{89}(29,\cdot)\) \(\chi_{89}(30,\cdot)\) \(\chi_{89}(31,\cdot)\) \(\chi_{89}(33,\cdot)\) \(\chi_{89}(35,\cdot)\) \(\chi_{89}(38,\cdot)\) \(\chi_{89}(41,\cdot)\) \(\chi_{89}(43,\cdot)\) \(\chi_{89}(46,\cdot)\) \(\chi_{89}(48,\cdot)\) \(\chi_{89}(51,\cdot)\) \(\chi_{89}(54,\cdot)\) \(\chi_{89}(56,\cdot)\) \(\chi_{89}(58,\cdot)\) \(\chi_{89}(59,\cdot)\) \(\chi_{89}(60,\cdot)\) \(\chi_{89}(61,\cdot)\) \(\chi_{89}(62,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{69}{88}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{69}{88}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{29}{88}\right)\)\(e\left(\frac{45}{88}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{19}{22}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{88})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 89 }(61,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{89}(61,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(61,r) e\left(\frac{2r}{89}\right) = 1.772711076+9.2659319791i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 89 }(61,·),\chi_{ 89 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{89}(61,\cdot),\chi_{89}(1,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(61,r) \chi_{89}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 89 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{89}(61,·)) = \sum_{r \in \Z/89\Z} \chi_{89}(61,r) e\left(\frac{1 r + 2 r^{-1}}{89}\right) = -6.7521247177+-0.9708089727i \)