from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,40,24]))
pari: [g,chi] = znchar(Mod(75,8512))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.js
\(\chi_{8512}(75,\cdot)\) \(\chi_{8512}(227,\cdot)\) \(\chi_{8512}(1139,\cdot)\) \(\chi_{8512}(1291,\cdot)\) \(\chi_{8512}(2203,\cdot)\) \(\chi_{8512}(2355,\cdot)\) \(\chi_{8512}(3267,\cdot)\) \(\chi_{8512}(3419,\cdot)\) \(\chi_{8512}(4331,\cdot)\) \(\chi_{8512}(4483,\cdot)\) \(\chi_{8512}(5395,\cdot)\) \(\chi_{8512}(5547,\cdot)\) \(\chi_{8512}(6459,\cdot)\) \(\chi_{8512}(6611,\cdot)\) \(\chi_{8512}(7523,\cdot)\) \(\chi_{8512}(7675,\cdot)\)
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
\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{5}{6}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(75, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)