from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,4,6,16,0]))
pari: [g,chi] = znchar(Mod(38567,221760))
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(10080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{10080}(7067,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 221760.bvd
\(\chi_{221760}(23,\cdot)\) \(\chi_{221760}(5303,\cdot)\) \(\chi_{221760}(33287,\cdot)\) \(\chi_{221760}(38567,\cdot)\) \(\chi_{221760}(110903,\cdot)\) \(\chi_{221760}(116183,\cdot)\) \(\chi_{221760}(144167,\cdot)\) \(\chi_{221760}(149447,\cdot)\)
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: | Number field defined by a degree 24 polynomial |
Values on generators
\((48511,124741,98561,133057,190081,141121)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{6}\right),i,e\left(\frac{2}{3}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 221760 }(38567, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(-1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)