Properties

Conductor 25
Order 20
Real No
Primitive No
Parity Odd
Orbit Label 50.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(50)
sage: chi = H[23]
pari: [g,chi] = znchar(Mod(23,50))

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 = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 50.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 ]

\(\chi_{50}(3,\cdot)\) \(\chi_{50}(13,\cdot)\) \(\chi_{50}(17,\cdot)\) \(\chi_{50}(23,\cdot)\) \(\chi_{50}(27,\cdot)\) \(\chi_{50}(33,\cdot)\) \(\chi_{50}(37,\cdot)\) \(\chi_{50}(47,\cdot)\)

Inducing primitive character

\(\chi_{25}(23,\cdot)\)

Values on generators

\(27\) → \(e\left(\frac{11}{20}\right)\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(-i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{20}\right)\)
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)
\( \tau_{ a }( \chi_{ 50 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{50}(23,\cdot)) = \sum_{r\in \Z/50\Z} \chi_{50}(23,r) e\left(\frac{r}{25}\right) = 4.9114362536+-0.9369065729i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 50 }(23,·),\chi_{ 50 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{50}(23,\cdot),\chi_{50}(1,\cdot)) = \sum_{r\in \Z/50\Z} \chi_{50}(23,r) \chi_{50}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 50 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{50}(23,·)) = \sum_{r \in \Z/50\Z} \chi_{50}(23,r) e\left(\frac{1 r + 2 r^{-1}}{50}\right) = -8.5014486433+-1.3464971853i \)