from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(86190, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,36,40,45]))
pari: [g,chi] = znchar(Mod(23,86190))
Basic properties
| Modulus: | \(86190\) | |
| Conductor: | \(3315\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3315}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 86190.ka
\(\chi_{86190}(23,\cdot)\) \(\chi_{86190}(5093,\cdot)\) \(\chi_{86190}(8597,\cdot)\) \(\chi_{86190}(16247,\cdot)\) \(\chi_{86190}(18737,\cdot)\) \(\chi_{86190}(26387,\cdot)\) \(\chi_{86190}(54227,\cdot)\) \(\chi_{86190}(58283,\cdot)\) \(\chi_{86190}(61877,\cdot)\) \(\chi_{86190}(63353,\cdot)\) \(\chi_{86190}(64367,\cdot)\) \(\chi_{86190}(65933,\cdot)\) \(\chi_{86190}(71003,\cdot)\) \(\chi_{86190}(72017,\cdot)\) \(\chi_{86190}(78563,\cdot)\) \(\chi_{86190}(83633,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((57461,34477,57631,45631)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 86190 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)