Properties

Conductor 41
Order 40
Real No
Primitive No
Parity Odd
Orbit Label 82.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(82)
sage: chi = H[75]
pari: [g,chi] = znchar(Mod(75,82))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 41
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 40
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 82.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 ]

\(\chi_{82}(7,\cdot)\) \(\chi_{82}(11,\cdot)\) \(\chi_{82}(13,\cdot)\) \(\chi_{82}(15,\cdot)\) \(\chi_{82}(17,\cdot)\) \(\chi_{82}(19,\cdot)\) \(\chi_{82}(29,\cdot)\) \(\chi_{82}(35,\cdot)\) \(\chi_{82}(47,\cdot)\) \(\chi_{82}(53,\cdot)\) \(\chi_{82}(63,\cdot)\) \(\chi_{82}(65,\cdot)\) \(\chi_{82}(67,\cdot)\) \(\chi_{82}(69,\cdot)\) \(\chi_{82}(71,\cdot)\) \(\chi_{82}(75,\cdot)\)

Inducing primitive character

\(\chi_{41}(34,\cdot)\)

Values on generators

\(47\) → \(e\left(\frac{19}{40}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{21}{40}\right)\)\(i\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{13}{20}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{40})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 82 }(75,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{82}(75,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(75,r) e\left(\frac{r}{41}\right) = 3.4436734695+-5.3982509237i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 82 }(75,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{82}(75,\cdot),\chi_{82}(1,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(75,r) \chi_{82}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 82 }(75,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{82}(75,·)) = \sum_{r \in \Z/82\Z} \chi_{82}(75,r) e\left(\frac{1 r + 2 r^{-1}}{82}\right) = 2.7531781292+-1.4028143238i \)