from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,20,8]))
pari: [g,chi] = znchar(Mod(7607,8512))
Basic properties
| Modulus: | \(8512\) | |
| Conductor: | \(4256\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4256}(2819,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.hi
\(\chi_{8512}(87,\cdot)\) \(\chi_{8512}(1223,\cdot)\) \(\chi_{8512}(2215,\cdot)\) \(\chi_{8512}(3351,\cdot)\) \(\chi_{8512}(4343,\cdot)\) \(\chi_{8512}(5479,\cdot)\) \(\chi_{8512}(6471,\cdot)\) \(\chi_{8512}(7607,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(7607, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)