Properties

Conductor 83
Order 82
Real No
Primitive Yes
Parity Odd
Orbit Label 83.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(83)
sage: chi = H[74]
pari: [g,chi] = znchar(Mod(74,83))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 83
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 82
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 = 83.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 ]

\(\chi_{83}(2,\cdot)\) \(\chi_{83}(5,\cdot)\) \(\chi_{83}(6,\cdot)\) \(\chi_{83}(8,\cdot)\) \(\chi_{83}(13,\cdot)\) \(\chi_{83}(14,\cdot)\) \(\chi_{83}(15,\cdot)\) \(\chi_{83}(18,\cdot)\) \(\chi_{83}(19,\cdot)\) \(\chi_{83}(20,\cdot)\) \(\chi_{83}(22,\cdot)\) \(\chi_{83}(24,\cdot)\) \(\chi_{83}(32,\cdot)\) \(\chi_{83}(34,\cdot)\) \(\chi_{83}(35,\cdot)\) \(\chi_{83}(39,\cdot)\) \(\chi_{83}(42,\cdot)\) \(\chi_{83}(43,\cdot)\) \(\chi_{83}(45,\cdot)\) \(\chi_{83}(46,\cdot)\) \(\chi_{83}(47,\cdot)\) \(\chi_{83}(50,\cdot)\) \(\chi_{83}(52,\cdot)\) \(\chi_{83}(53,\cdot)\) \(\chi_{83}(54,\cdot)\) \(\chi_{83}(55,\cdot)\) \(\chi_{83}(56,\cdot)\) \(\chi_{83}(57,\cdot)\) \(\chi_{83}(58,\cdot)\) \(\chi_{83}(60,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{21}{82}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{21}{82}\right)\)\(e\left(\frac{18}{41}\right)\)\(e\left(\frac{21}{41}\right)\)\(e\left(\frac{75}{82}\right)\)\(e\left(\frac{57}{82}\right)\)\(e\left(\frac{2}{41}\right)\)\(e\left(\frac{63}{82}\right)\)\(e\left(\frac{36}{41}\right)\)\(e\left(\frac{7}{41}\right)\)\(e\left(\frac{6}{41}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{41})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 83 }(74,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{83}(74,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(74,r) e\left(\frac{2r}{83}\right) = -0.1832343828+9.1085907341i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 83 }(74,·),\chi_{ 83 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{83}(74,\cdot),\chi_{83}(1,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(74,r) \chi_{83}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 83 }(74,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{83}(74,·)) = \sum_{r \in \Z/83\Z} \chi_{83}(74,r) e\left(\frac{1 r + 2 r^{-1}}{83}\right) = 9.822569344+-9.4532750786i \)