from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,32,32]))
pari: [g,chi] = znchar(Mod(3469,8512))
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.jw
\(\chi_{8512}(277,\cdot)\) \(\chi_{8512}(653,\cdot)\) \(\chi_{8512}(1341,\cdot)\) \(\chi_{8512}(1717,\cdot)\) \(\chi_{8512}(2405,\cdot)\) \(\chi_{8512}(2781,\cdot)\) \(\chi_{8512}(3469,\cdot)\) \(\chi_{8512}(3845,\cdot)\) \(\chi_{8512}(4533,\cdot)\) \(\chi_{8512}(4909,\cdot)\) \(\chi_{8512}(5597,\cdot)\) \(\chi_{8512}(5973,\cdot)\) \(\chi_{8512}(6661,\cdot)\) \(\chi_{8512}(7037,\cdot)\) \(\chi_{8512}(7725,\cdot)\) \(\chi_{8512}(8101,\cdot)\)
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
\((5055,6917,7297,3137)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(3469, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)