from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(629, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
pari: [g,chi] = znchar(Mod(528,629))
Basic properties
Modulus: | \(629\) | |
Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(3\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{37}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 629.e
\(\chi_{629}(137,\cdot)\) \(\chi_{629}(528,\cdot)\)
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: | \(\mathbb{Q}(\zeta_3)\) |
Fixed field: | 3.3.1369.1 |
Values on generators
\((445,409)\) → \((1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 629 }(528, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(1\) |
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)