from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,32,0]))
pari: [g,chi] = znchar(Mod(133,6336))
Basic properties
| Modulus: | \(6336\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(133,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6336.ev
\(\chi_{6336}(133,\cdot)\) \(\chi_{6336}(661,\cdot)\) \(\chi_{6336}(925,\cdot)\) \(\chi_{6336}(1453,\cdot)\) \(\chi_{6336}(1717,\cdot)\) \(\chi_{6336}(2245,\cdot)\) \(\chi_{6336}(2509,\cdot)\) \(\chi_{6336}(3037,\cdot)\) \(\chi_{6336}(3301,\cdot)\) \(\chi_{6336}(3829,\cdot)\) \(\chi_{6336}(4093,\cdot)\) \(\chi_{6336}(4621,\cdot)\) \(\chi_{6336}(4885,\cdot)\) \(\chi_{6336}(5413,\cdot)\) \(\chi_{6336}(5677,\cdot)\) \(\chi_{6336}(6205,\cdot)\)
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
\((4159,4357,3521,1729)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{2}{3}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)