from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4641, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,40,36,15]))
pari: [g,chi] = znchar(Mod(5,4641))
Basic properties
| Modulus: | \(4641\) | |
| Conductor: | \(4641\) | 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 4641.nm
\(\chi_{4641}(5,\cdot)\) \(\chi_{4641}(122,\cdot)\) \(\chi_{4641}(164,\cdot)\) \(\chi_{4641}(668,\cdot)\) \(\chi_{4641}(1916,\cdot)\) \(\chi_{4641}(2462,\cdot)\) \(\chi_{4641}(2579,\cdot)\) \(\chi_{4641}(2777,\cdot)\) \(\chi_{4641}(3050,\cdot)\) \(\chi_{4641}(3125,\cdot)\) \(\chi_{4641}(3440,\cdot)\) \(\chi_{4641}(3713,\cdot)\) \(\chi_{4641}(3869,\cdot)\) \(\chi_{4641}(4100,\cdot)\) \(\chi_{4641}(4142,\cdot)\) \(\chi_{4641}(4532,\cdot)\)
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
\((3095,3979,3928,547)\) → \((-1,e\left(\frac{5}{6}\right),-i,e\left(\frac{5}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(19\) | \(20\) | \(22\) |
| \( \chi_{ 4641 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)