from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,56]))
pari: [g,chi] = znchar(Mod(127,3484))
Basic properties
| Modulus: | \(3484\) | |
| Conductor: | \(3484\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3484.dh
\(\chi_{3484}(127,\cdot)\) \(\chi_{3484}(199,\cdot)\) \(\chi_{3484}(719,\cdot)\) \(\chi_{3484}(823,\cdot)\) \(\chi_{3484}(907,\cdot)\) \(\chi_{3484}(959,\cdot)\) \(\chi_{3484}(1375,\cdot)\) \(\chi_{3484}(1551,\cdot)\) \(\chi_{3484}(1759,\cdot)\) \(\chi_{3484}(1863,\cdot)\) \(\chi_{3484}(1915,\cdot)\) \(\chi_{3484}(1947,\cdot)\) \(\chi_{3484}(1999,\cdot)\) \(\chi_{3484}(2103,\cdot)\) \(\chi_{3484}(2227,\cdot)\) \(\chi_{3484}(2311,\cdot)\) \(\chi_{3484}(2435,\cdot)\) \(\chi_{3484}(2467,\cdot)\) \(\chi_{3484}(2727,\cdot)\) \(\chi_{3484}(3423,\cdot)\)
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
\((1743,1341,3017)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{28}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3484 }(127, a) \) | \(-1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)