from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24704, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,9,10]))
pari: [g,chi] = znchar(Mod(27,24704))
Basic properties
| Modulus: | \(24704\) | |
| Conductor: | \(24704\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | 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 24704.hs
\(\chi_{24704}(3,\cdot)\) \(\chi_{24704}(27,\cdot)\) \(\chi_{24704}(243,\cdot)\) \(\chi_{24704}(915,\cdot)\) \(\chi_{24704}(2187,\cdot)\) \(\chi_{24704}(4219,\cdot)\) \(\chi_{24704}(7331,\cdot)\) \(\chi_{24704}(8235,\cdot)\) \(\chi_{24704}(12355,\cdot)\) \(\chi_{24704}(12379,\cdot)\) \(\chi_{24704}(12595,\cdot)\) \(\chi_{24704}(13267,\cdot)\) \(\chi_{24704}(14539,\cdot)\) \(\chi_{24704}(16571,\cdot)\) \(\chi_{24704}(19683,\cdot)\) \(\chi_{24704}(20587,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((24319,773,23937)\) → \((-1,e\left(\frac{9}{32}\right),e\left(\frac{5}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 24704 }(27, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) |
sage: chi.jacobi_sum(n)