from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1547, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([16,40,45]))
pari: [g,chi] = znchar(Mod(23,1547))
Basic properties
| Modulus: | \(1547\) | |
| Conductor: | \(1547\) | 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 1547.fy
\(\chi_{1547}(23,\cdot)\) \(\chi_{1547}(95,\cdot)\) \(\chi_{1547}(114,\cdot)\) \(\chi_{1547}(277,\cdot)\) \(\chi_{1547}(296,\cdot)\) \(\chi_{1547}(368,\cdot)\) \(\chi_{1547}(550,\cdot)\) \(\chi_{1547}(641,\cdot)\) \(\chi_{1547}(660,\cdot)\) \(\chi_{1547}(751,\cdot)\) \(\chi_{1547}(823,\cdot)\) \(\chi_{1547}(1115,\cdot)\) \(\chi_{1547}(1187,\cdot)\) \(\chi_{1547}(1278,\cdot)\) \(\chi_{1547}(1297,\cdot)\) \(\chi_{1547}(1388,\cdot)\)
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
\((885,834,547)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1547 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(i\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) |
sage: chi.jacobi_sum(n)