from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,15,4]))
pari: [g,chi] = znchar(Mod(1073,1152))
Basic properties
| Modulus: | \(1152\) | |
| Conductor: | \(288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{288}(245,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.bi
\(\chi_{1152}(113,\cdot)\) \(\chi_{1152}(209,\cdot)\) \(\chi_{1152}(401,\cdot)\) \(\chi_{1152}(497,\cdot)\) \(\chi_{1152}(689,\cdot)\) \(\chi_{1152}(785,\cdot)\) \(\chi_{1152}(977,\cdot)\) \(\chi_{1152}(1073,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.0.1486465269728735333725176976133731985582456832.1 |
Values on generators
\((127,901,641)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1152 }(1073, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)