from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,16]))
pari: [g,chi] = znchar(Mod(103,1152))
Basic properties
| Modulus: | \(1152\) | |
| 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 \(\chi_{576}(571,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.bo
\(\chi_{1152}(7,\cdot)\) \(\chi_{1152}(103,\cdot)\) \(\chi_{1152}(151,\cdot)\) \(\chi_{1152}(247,\cdot)\) \(\chi_{1152}(295,\cdot)\) \(\chi_{1152}(391,\cdot)\) \(\chi_{1152}(439,\cdot)\) \(\chi_{1152}(535,\cdot)\) \(\chi_{1152}(583,\cdot)\) \(\chi_{1152}(679,\cdot)\) \(\chi_{1152}(727,\cdot)\) \(\chi_{1152}(823,\cdot)\) \(\chi_{1152}(871,\cdot)\) \(\chi_{1152}(967,\cdot)\) \(\chi_{1152}(1015,\cdot)\) \(\chi_{1152}(1111,\cdot)\)
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
\((127,901,641)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1152 }(103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)