Properties

Conductor 41
Order 20
Real No
Primitive No
Parity Even
Orbit Label 82.g

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[21]
pari: [g,chi] = znchar(Mod(21,82))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 41
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 20
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 = Even
Orbit label = 82.g
Orbit index = 7

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 ]

\(\chi_{82}(5,\cdot)\) \(\chi_{82}(21,\cdot)\) \(\chi_{82}(33,\cdot)\) \(\chi_{82}(39,\cdot)\) \(\chi_{82}(43,\cdot)\) \(\chi_{82}(49,\cdot)\) \(\chi_{82}(61,\cdot)\) \(\chi_{82}(77,\cdot)\)

Inducing primitive character

\(\chi_{41}(21,\cdot)\)

Values on generators

\(47\) → \(e\left(\frac{7}{20}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(-1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 82 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{82}(21,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(21,r) e\left(\frac{r}{41}\right) = 1.9046811409+6.1132797868i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 82 }(21,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{82}(21,\cdot),\chi_{82}(1,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(21,r) \chi_{82}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 82 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{82}(21,·)) = \sum_{r \in \Z/82\Z} \chi_{82}(21,r) e\left(\frac{1 r + 2 r^{-1}}{82}\right) = -9.1382727711+-2.9692048129i \)