from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(764, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,3]))
pari: [g,chi] = znchar(Mod(37,764))
Basic properties
| Modulus: | \(764\) | |
| Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{191}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 764.j
\(\chi_{764}(37,\cdot)\) \(\chi_{764}(41,\cdot)\) \(\chi_{764}(161,\cdot)\) \(\chi_{764}(185,\cdot)\) \(\chi_{764}(205,\cdot)\) \(\chi_{764}(229,\cdot)\) \(\chi_{764}(257,\cdot)\) \(\chi_{764}(261,\cdot)\) \(\chi_{764}(313,\cdot)\) \(\chi_{764}(357,\cdot)\) \(\chi_{764}(377,\cdot)\) \(\chi_{764}(393,\cdot)\) \(\chi_{764}(413,\cdot)\) \(\chi_{764}(437,\cdot)\) \(\chi_{764}(521,\cdot)\) \(\chi_{764}(537,\cdot)\) \(\chi_{764}(541,\cdot)\) \(\chi_{764}(657,\cdot)\)
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 38 polynomial |
Values on generators
\((383,401)\) → \((1,e\left(\frac{3}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 764 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(-1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{25}{38}\right)\) |
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)