from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1547, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,32,15]))
pari: [g,chi] = znchar(Mod(22,1547))
Basic properties
| Modulus: | \(1547\) | |
| Conductor: | \(221\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{221}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1547.gx
\(\chi_{1547}(22,\cdot)\) \(\chi_{1547}(29,\cdot)\) \(\chi_{1547}(113,\cdot)\) \(\chi_{1547}(211,\cdot)\) \(\chi_{1547}(295,\cdot)\) \(\chi_{1547}(386,\cdot)\) \(\chi_{1547}(568,\cdot)\) \(\chi_{1547}(575,\cdot)\) \(\chi_{1547}(666,\cdot)\) \(\chi_{1547}(932,\cdot)\) \(\chi_{1547}(1023,\cdot)\) \(\chi_{1547}(1030,\cdot)\) \(\chi_{1547}(1212,\cdot)\) \(\chi_{1547}(1303,\cdot)\) \(\chi_{1547}(1387,\cdot)\) \(\chi_{1547}(1485,\cdot)\)
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
\((885,834,547)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{5}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1547 }(22, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)