from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,4,0]))
pari: [g,chi] = znchar(Mod(7031,8512))
Basic properties
| Modulus: | \(8512\) | |
| Conductor: | \(224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{224}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.if
\(\chi_{8512}(647,\cdot)\) \(\chi_{8512}(1559,\cdot)\) \(\chi_{8512}(2775,\cdot)\) \(\chi_{8512}(3687,\cdot)\) \(\chi_{8512}(4903,\cdot)\) \(\chi_{8512}(5815,\cdot)\) \(\chi_{8512}(7031,\cdot)\) \(\chi_{8512}(7943,\cdot)\)
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: | 24.24.790224330201082600125157415256880139617697792.1 |
Values on generators
\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(7031, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)