from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,32]))
pari: [g,chi] = znchar(Mod(361,6272))
Basic properties
| Modulus: | \(6272\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6272.cl
\(\chi_{6272}(361,\cdot)\) \(\chi_{6272}(569,\cdot)\) \(\chi_{6272}(1145,\cdot)\) \(\chi_{6272}(1353,\cdot)\) \(\chi_{6272}(1929,\cdot)\) \(\chi_{6272}(2137,\cdot)\) \(\chi_{6272}(2713,\cdot)\) \(\chi_{6272}(2921,\cdot)\) \(\chi_{6272}(3497,\cdot)\) \(\chi_{6272}(3705,\cdot)\) \(\chi_{6272}(4281,\cdot)\) \(\chi_{6272}(4489,\cdot)\) \(\chi_{6272}(5065,\cdot)\) \(\chi_{6272}(5273,\cdot)\) \(\chi_{6272}(5849,\cdot)\) \(\chi_{6272}(6057,\cdot)\)
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
\((4607,3333,4609)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 6272 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) |
sage: chi.jacobi_sum(n)