from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(723, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,3]))
pari: [g,chi] = znchar(Mod(266,723))
Basic properties
Modulus: | \(723\) | |
Conductor: | \(723\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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 723.w
\(\chi_{723}(266,\cdot)\) \(\chi_{723}(281,\cdot)\) \(\chi_{723}(347,\cdot)\) \(\chi_{723}(476,\cdot)\) \(\chi_{723}(488,\cdot)\) \(\chi_{723}(617,\cdot)\) \(\chi_{723}(683,\cdot)\) \(\chi_{723}(698,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.0.107037598360768458712497472964188759976745670736889.1 |
Values on generators
\((242,7)\) → \((-1,e\left(\frac{3}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 723 }(266, a) \) | \(-1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) |
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)