from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,16,12,32,0]))
pari: [g,chi] = znchar(Mod(67,221760))
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(20160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{20160}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 221760.ctz
\(\chi_{221760}(67,\cdot)\) \(\chi_{221760}(44683,\cdot)\) \(\chi_{221760}(49963,\cdot)\) \(\chi_{221760}(50227,\cdot)\) \(\chi_{221760}(55507,\cdot)\) \(\chi_{221760}(100123,\cdot)\) \(\chi_{221760}(105403,\cdot)\) \(\chi_{221760}(105667,\cdot)\) \(\chi_{221760}(110947,\cdot)\) \(\chi_{221760}(155563,\cdot)\) \(\chi_{221760}(160843,\cdot)\) \(\chi_{221760}(161107,\cdot)\) \(\chi_{221760}(166387,\cdot)\) \(\chi_{221760}(211003,\cdot)\) \(\chi_{221760}(216283,\cdot)\) \(\chi_{221760}(216547,\cdot)\)
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
\((48511,124741,98561,133057,190081,141121)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right),i,e\left(\frac{2}{3}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 221760 }(67, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)