from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,40]))
pari: [g,chi] = znchar(Mod(95,4608))
Basic properties
| Modulus: | \(4608\) | |
| 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 \(\chi_{576}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4608.bp
\(\chi_{4608}(95,\cdot)\) \(\chi_{4608}(479,\cdot)\) \(\chi_{4608}(671,\cdot)\) \(\chi_{4608}(1055,\cdot)\) \(\chi_{4608}(1247,\cdot)\) \(\chi_{4608}(1631,\cdot)\) \(\chi_{4608}(1823,\cdot)\) \(\chi_{4608}(2207,\cdot)\) \(\chi_{4608}(2399,\cdot)\) \(\chi_{4608}(2783,\cdot)\) \(\chi_{4608}(2975,\cdot)\) \(\chi_{4608}(3359,\cdot)\) \(\chi_{4608}(3551,\cdot)\) \(\chi_{4608}(3935,\cdot)\) \(\chi_{4608}(4127,\cdot)\) \(\chi_{4608}(4511,\cdot)\)
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
\((3583,2053,4097)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 4608 }(95, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)