from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,36]))
pari: [g,chi] = znchar(Mod(1749,1805))
Basic properties
| Modulus: | \(1805\) | |
| Conductor: | \(1805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1805.v
\(\chi_{1805}(39,\cdot)\) \(\chi_{1805}(134,\cdot)\) \(\chi_{1805}(229,\cdot)\) \(\chi_{1805}(324,\cdot)\) \(\chi_{1805}(419,\cdot)\) \(\chi_{1805}(514,\cdot)\) \(\chi_{1805}(609,\cdot)\) \(\chi_{1805}(704,\cdot)\) \(\chi_{1805}(799,\cdot)\) \(\chi_{1805}(894,\cdot)\) \(\chi_{1805}(989,\cdot)\) \(\chi_{1805}(1179,\cdot)\) \(\chi_{1805}(1274,\cdot)\) \(\chi_{1805}(1369,\cdot)\) \(\chi_{1805}(1464,\cdot)\) \(\chi_{1805}(1559,\cdot)\) \(\chi_{1805}(1654,\cdot)\) \(\chi_{1805}(1749,\cdot)\)
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
\((362,1446)\) → \((-1,e\left(\frac{18}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1805 }(1749, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{7}{38}\right)\) |
sage: chi.jacobi_sum(n)