from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,32,24]))
pari: [g,chi] = znchar(Mod(4939,8512))
Basic properties
| Modulus: | \(8512\) | |
| Conductor: | \(8512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 8512.kb
\(\chi_{8512}(683,\cdot)\) \(\chi_{8512}(835,\cdot)\) \(\chi_{8512}(1747,\cdot)\) \(\chi_{8512}(1899,\cdot)\) \(\chi_{8512}(2811,\cdot)\) \(\chi_{8512}(2963,\cdot)\) \(\chi_{8512}(3875,\cdot)\) \(\chi_{8512}(4027,\cdot)\) \(\chi_{8512}(4939,\cdot)\) \(\chi_{8512}(5091,\cdot)\) \(\chi_{8512}(6003,\cdot)\) \(\chi_{8512}(6155,\cdot)\) \(\chi_{8512}(7067,\cdot)\) \(\chi_{8512}(7219,\cdot)\) \(\chi_{8512}(8131,\cdot)\) \(\chi_{8512}(8283,\cdot)\)
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
\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{2}{3}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(4939, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)