from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,0,8,39]))
pari: [g,chi] = znchar(Mod(335,1224))
Basic properties
| Modulus: | \(1224\) | |
| Conductor: | \(612\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{612}(335,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1224.cy
\(\chi_{1224}(23,\cdot)\) \(\chi_{1224}(95,\cdot)\) \(\chi_{1224}(167,\cdot)\) \(\chi_{1224}(311,\cdot)\) \(\chi_{1224}(335,\cdot)\) \(\chi_{1224}(479,\cdot)\) \(\chi_{1224}(551,\cdot)\) \(\chi_{1224}(623,\cdot)\) \(\chi_{1224}(743,\cdot)\) \(\chi_{1224}(839,\cdot)\) \(\chi_{1224}(887,\cdot)\) \(\chi_{1224}(911,\cdot)\) \(\chi_{1224}(959,\cdot)\) \(\chi_{1224}(983,\cdot)\) \(\chi_{1224}(1031,\cdot)\) \(\chi_{1224}(1127,\cdot)\)
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
\((919,613,137,649)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1224 }(335, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)