from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6762, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,9]))
pari: [g,chi] = znchar(Mod(263,6762))
Basic properties
| Modulus: | \(6762\) | |
| Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{483}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6762.bv
\(\chi_{6762}(263,\cdot)\) \(\chi_{6762}(557,\cdot)\) \(\chi_{6762}(569,\cdot)\) \(\chi_{6762}(1157,\cdot)\) \(\chi_{6762}(1745,\cdot)\) \(\chi_{6762}(2039,\cdot)\) \(\chi_{6762}(2321,\cdot)\) \(\chi_{6762}(2333,\cdot)\) \(\chi_{6762}(2627,\cdot)\) \(\chi_{6762}(2909,\cdot)\) \(\chi_{6762}(3791,\cdot)\) \(\chi_{6762}(4085,\cdot)\) \(\chi_{6762}(4391,\cdot)\) \(\chi_{6762}(4979,\cdot)\) \(\chi_{6762}(5261,\cdot)\) \(\chi_{6762}(5849,\cdot)\) \(\chi_{6762}(5861,\cdot)\) \(\chi_{6762}(6155,\cdot)\) \(\chi_{6762}(6437,\cdot)\) \(\chi_{6762}(6731,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2255,3727,3823)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6762 }(263, a) \) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)