from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,32]))
pari: [g,chi] = znchar(Mod(49,4864))
Basic properties
| Modulus: | \(4864\) | |
| Conductor: | \(1216\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1216}(885,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4864.cb
\(\chi_{4864}(49,\cdot)\) \(\chi_{4864}(273,\cdot)\) \(\chi_{4864}(657,\cdot)\) \(\chi_{4864}(881,\cdot)\) \(\chi_{4864}(1265,\cdot)\) \(\chi_{4864}(1489,\cdot)\) \(\chi_{4864}(1873,\cdot)\) \(\chi_{4864}(2097,\cdot)\) \(\chi_{4864}(2481,\cdot)\) \(\chi_{4864}(2705,\cdot)\) \(\chi_{4864}(3089,\cdot)\) \(\chi_{4864}(3313,\cdot)\) \(\chi_{4864}(3697,\cdot)\) \(\chi_{4864}(3921,\cdot)\) \(\chi_{4864}(4305,\cdot)\) \(\chi_{4864}(4529,\cdot)\)
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
\((3839,2053,4353)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) |
sage: chi.jacobi_sum(n)