from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(764, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,8]))
pari: [g,chi] = znchar(Mod(709,764))
Basic properties
Modulus: | \(764\) | |
Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{191}(136,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 764.i
\(\chi_{764}(5,\cdot)\) \(\chi_{764}(25,\cdot)\) \(\chi_{764}(69,\cdot)\) \(\chi_{764}(121,\cdot)\) \(\chi_{764}(125,\cdot)\) \(\chi_{764}(153,\cdot)\) \(\chi_{764}(177,\cdot)\) \(\chi_{764}(197,\cdot)\) \(\chi_{764}(221,\cdot)\) \(\chi_{764}(341,\cdot)\) \(\chi_{764}(345,\cdot)\) \(\chi_{764}(489,\cdot)\) \(\chi_{764}(605,\cdot)\) \(\chi_{764}(609,\cdot)\) \(\chi_{764}(625,\cdot)\) \(\chi_{764}(709,\cdot)\) \(\chi_{764}(733,\cdot)\) \(\chi_{764}(753,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 19 polynomial |
Values on generators
\((383,401)\) → \((1,e\left(\frac{4}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 764 }(709, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
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)