Properties

Conductor 25
Order 20
Real No
Primitive Yes
Parity Odd
Orbit Label 25.f

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(25)
sage: chi = H[2]
pari: [g,chi] = znchar(Mod(2,25))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 25
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 = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 25.f
Orbit index = 6

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 ]

\(\chi_{25}(2,\cdot)\) \(\chi_{25}(3,\cdot)\) \(\chi_{25}(8,\cdot)\) \(\chi_{25}(12,\cdot)\) \(\chi_{25}(13,\cdot)\) \(\chi_{25}(17,\cdot)\) \(\chi_{25}(22,\cdot)\) \(\chi_{25}(23,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{1}{20}\right)\)

Values

-112346789111213
\(-1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{19}{20}\right)\)
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)
\( \tau_{ a }( \chi_{ 25 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{25}(2,\cdot)) = \sum_{r\in \Z/25\Z} \chi_{25}(2,r) e\left(\frac{2r}{25}\right) = -4.3815334002+2.4087683705i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 25 }(2,·),\chi_{ 25 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{25}(2,\cdot),\chi_{25}(1,\cdot)) = \sum_{r\in \Z/25\Z} \chi_{25}(2,r) \chi_{25}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 25 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{25}(2,·)) = \sum_{r \in \Z/25\Z} \chi_{25}(2,r) e\left(\frac{1 r + 2 r^{-1}}{25}\right) = -0.0 \)