from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(633, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,38]))
pari: [g,chi] = znchar(Mod(101,633))
Basic properties
Modulus: | \(633\) | |
Conductor: | \(633\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 633.x
\(\chi_{633}(101,\cdot)\) \(\chi_{633}(161,\cdot)\) \(\chi_{633}(173,\cdot)\) \(\chi_{633}(179,\cdot)\) \(\chi_{633}(185,\cdot)\) \(\chi_{633}(245,\cdot)\) \(\chi_{633}(254,\cdot)\) \(\chi_{633}(284,\cdot)\) \(\chi_{633}(389,\cdot)\) \(\chi_{633}(476,\cdot)\) \(\chi_{633}(539,\cdot)\) \(\chi_{633}(602,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{21})\) |
Fixed field: | 42.0.9791394774306703661413185563624237182966995193131573802248957732013305671562342253106532979233656192403.1 |
Values on generators
\((212,424)\) → \((-1,e\left(\frac{19}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 633 }(101, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)