Properties

Conductor 61
Order 5
Real No
Primitive Yes
Parity Even
Orbit Label 61.e

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(61)
sage: chi = H[20]
pari: [g,chi] = znchar(Mod(20,61))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 61
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 5
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 = 61.e
Orbit index = 5

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 ]

\(\chi_{61}(9,\cdot)\) \(\chi_{61}(20,\cdot)\) \(\chi_{61}(34,\cdot)\) \(\chi_{61}(58,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{2}{5}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 61 }(20,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(20,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(20,r) e\left(\frac{2r}{61}\right) = -7.290629108+-2.8012010298i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 61 }(20,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(20,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(20,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 61 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(20,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(20,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = 4.5109477404+13.8832695977i \)