from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,16]))
pari: [g,chi] = znchar(Mod(1375,2304))
Basic properties
| Modulus: | \(2304\) | |
| 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 \(\chi_{288}(259,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2304.bh
\(\chi_{2304}(31,\cdot)\) \(\chi_{2304}(223,\cdot)\) \(\chi_{2304}(607,\cdot)\) \(\chi_{2304}(799,\cdot)\) \(\chi_{2304}(1183,\cdot)\) \(\chi_{2304}(1375,\cdot)\) \(\chi_{2304}(1759,\cdot)\) \(\chi_{2304}(1951,\cdot)\)
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: | 24.0.18351423083070806589199715754737431920771072.1 |
Values on generators
\((1279,2053,1793)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2304 }(1375, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)