from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,4,9]))
pari: [g,chi] = znchar(Mod(418,8619))
Basic properties
| Modulus: | \(8619\) | |
| Conductor: | \(221\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{221}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8619.cy
\(\chi_{8619}(418,\cdot)\) \(\chi_{8619}(925,\cdot)\) \(\chi_{8619}(1540,\cdot)\) \(\chi_{8619}(2047,\cdot)\) \(\chi_{8619}(3361,\cdot)\) \(\chi_{8619}(3967,\cdot)\) \(\chi_{8619}(4483,\cdot)\) \(\chi_{8619}(4882,\cdot)\) \(\chi_{8619}(5089,\cdot)\) \(\chi_{8619}(5896,\cdot)\) \(\chi_{8619}(5995,\cdot)\) \(\chi_{8619}(6004,\cdot)\) \(\chi_{8619}(7018,\cdot)\) \(\chi_{8619}(7117,\cdot)\) \(\chi_{8619}(7417,\cdot)\) \(\chi_{8619}(8539,\cdot)\)
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
\((5747,5917,2536)\) → \((1,e\left(\frac{1}{12}\right),e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 8619 }(418, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) |
sage: chi.jacobi_sum(n)