from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3997, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,14]))
pari: [g,chi] = znchar(Mod(3989,3997))
Basic properties
| Modulus: | \(3997\) | |
| Conductor: | \(3997\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3997.bp
\(\chi_{3997}(55,\cdot)\) \(\chi_{3997}(741,\cdot)\) \(\chi_{3997}(1273,\cdot)\) \(\chi_{3997}(1413,\cdot)\) \(\chi_{3997}(1448,\cdot)\) \(\chi_{3997}(1532,\cdot)\) \(\chi_{3997}(1777,\cdot)\) \(\chi_{3997}(1812,\cdot)\) \(\chi_{3997}(2036,\cdot)\) \(\chi_{3997}(2120,\cdot)\) \(\chi_{3997}(2400,\cdot)\) \(\chi_{3997}(2498,\cdot)\) \(\chi_{3997}(3205,\cdot)\) \(\chi_{3997}(3457,\cdot)\) \(\chi_{3997}(3485,\cdot)\) \(\chi_{3997}(3520,\cdot)\) \(\chi_{3997}(3779,\cdot)\) \(\chi_{3997}(3989,\cdot)\)
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
\((2285,1716)\) → \((-1,e\left(\frac{7}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3997 }(3989, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) |
sage: chi.jacobi_sum(n)