from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3997, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,33]))
pari: [g,chi] = znchar(Mod(8,3997))
Basic properties
| Modulus: | \(3997\) | |
| Conductor: | \(571\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{571}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3997.br
\(\chi_{3997}(8,\cdot)\) \(\chi_{3997}(218,\cdot)\) \(\chi_{3997}(477,\cdot)\) \(\chi_{3997}(512,\cdot)\) \(\chi_{3997}(540,\cdot)\) \(\chi_{3997}(792,\cdot)\) \(\chi_{3997}(1499,\cdot)\) \(\chi_{3997}(1597,\cdot)\) \(\chi_{3997}(1877,\cdot)\) \(\chi_{3997}(1961,\cdot)\) \(\chi_{3997}(2185,\cdot)\) \(\chi_{3997}(2220,\cdot)\) \(\chi_{3997}(2465,\cdot)\) \(\chi_{3997}(2549,\cdot)\) \(\chi_{3997}(2584,\cdot)\) \(\chi_{3997}(2724,\cdot)\) \(\chi_{3997}(3256,\cdot)\) \(\chi_{3997}(3942,\cdot)\)
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
\((2285,1716)\) → \((1,e\left(\frac{33}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3997 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) |
sage: chi.jacobi_sum(n)