Properties

Conductor 83
Order 41
Real No
Primitive Yes
Parity Even
Orbit Label 83.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(83)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,83))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 83
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 41
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 = 83.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 ]

\(\chi_{83}(3,\cdot)\) \(\chi_{83}(4,\cdot)\) \(\chi_{83}(7,\cdot)\) \(\chi_{83}(9,\cdot)\) \(\chi_{83}(10,\cdot)\) \(\chi_{83}(11,\cdot)\) \(\chi_{83}(12,\cdot)\) \(\chi_{83}(16,\cdot)\) \(\chi_{83}(17,\cdot)\) \(\chi_{83}(21,\cdot)\) \(\chi_{83}(23,\cdot)\) \(\chi_{83}(25,\cdot)\) \(\chi_{83}(26,\cdot)\) \(\chi_{83}(27,\cdot)\) \(\chi_{83}(28,\cdot)\) \(\chi_{83}(29,\cdot)\) \(\chi_{83}(30,\cdot)\) \(\chi_{83}(31,\cdot)\) \(\chi_{83}(33,\cdot)\) \(\chi_{83}(36,\cdot)\) \(\chi_{83}(37,\cdot)\) \(\chi_{83}(38,\cdot)\) \(\chi_{83}(40,\cdot)\) \(\chi_{83}(41,\cdot)\) \(\chi_{83}(44,\cdot)\) \(\chi_{83}(48,\cdot)\) \(\chi_{83}(49,\cdot)\) \(\chi_{83}(51,\cdot)\) \(\chi_{83}(59,\cdot)\) \(\chi_{83}(61,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{12}{41}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{12}{41}\right)\)\(e\left(\frac{3}{41}\right)\)\(e\left(\frac{24}{41}\right)\)\(e\left(\frac{37}{41}\right)\)\(e\left(\frac{15}{41}\right)\)\(e\left(\frac{14}{41}\right)\)\(e\left(\frac{36}{41}\right)\)\(e\left(\frac{6}{41}\right)\)\(e\left(\frac{8}{41}\right)\)\(e\left(\frac{1}{41}\right)\)
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)
\( \tau_{ a }( \chi_{ 83 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{83}(11,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(11,r) e\left(\frac{2r}{83}\right) = 8.4637402002+-3.3712166682i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 83 }(11,·),\chi_{ 83 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{83}(11,\cdot),\chi_{83}(1,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(11,r) \chi_{83}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 83 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{83}(11,·)) = \sum_{r \in \Z/83\Z} \chi_{83}(11,r) e\left(\frac{1 r + 2 r^{-1}}{83}\right) = 3.4587296425+4.5374261431i \)