from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,8,0]))
pari: [g,chi] = znchar(Mod(251,8640))
Basic properties
| Modulus: | \(8640\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(443,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8640.gt
\(\chi_{8640}(251,\cdot)\) \(\chi_{8640}(611,\cdot)\) \(\chi_{8640}(1331,\cdot)\) \(\chi_{8640}(1691,\cdot)\) \(\chi_{8640}(2411,\cdot)\) \(\chi_{8640}(2771,\cdot)\) \(\chi_{8640}(3491,\cdot)\) \(\chi_{8640}(3851,\cdot)\) \(\chi_{8640}(4571,\cdot)\) \(\chi_{8640}(4931,\cdot)\) \(\chi_{8640}(5651,\cdot)\) \(\chi_{8640}(6011,\cdot)\) \(\chi_{8640}(6731,\cdot)\) \(\chi_{8640}(7091,\cdot)\) \(\chi_{8640}(7811,\cdot)\) \(\chi_{8640}(8171,\cdot)\)
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
\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) |
sage: chi.jacobi_sum(n)