from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(372, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,5,3]))
pari: [g,chi] = znchar(Mod(215,372))
Basic properties
Modulus: | \(372\) | |
Conductor: | \(372\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(10\) | 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 372.u
\(\chi_{372}(23,\cdot)\) \(\chi_{372}(215,\cdot)\) \(\chi_{372}(263,\cdot)\) \(\chi_{372}(275,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
Fixed field: | 10.0.6579024061484086272.1 |
Values on generators
\((187,125,313)\) → \((-1,-1,e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
\( \chi_{ 372 }(215, a) \) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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)