from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4356, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,11,15]))
pari: [g,chi] = znchar(Mod(3431,4356))
Basic properties
| Modulus: | \(4356\) | |
| Conductor: | \(4356\) | 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 4356.bs
\(\chi_{4356}(131,\cdot)\) \(\chi_{4356}(263,\cdot)\) \(\chi_{4356}(527,\cdot)\) \(\chi_{4356}(659,\cdot)\) \(\chi_{4356}(923,\cdot)\) \(\chi_{4356}(1055,\cdot)\) \(\chi_{4356}(1319,\cdot)\) \(\chi_{4356}(1715,\cdot)\) \(\chi_{4356}(1847,\cdot)\) \(\chi_{4356}(2111,\cdot)\) \(\chi_{4356}(2243,\cdot)\) \(\chi_{4356}(2507,\cdot)\) \(\chi_{4356}(2639,\cdot)\) \(\chi_{4356}(3035,\cdot)\) \(\chi_{4356}(3299,\cdot)\) \(\chi_{4356}(3431,\cdot)\) \(\chi_{4356}(3695,\cdot)\) \(\chi_{4356}(3827,\cdot)\) \(\chi_{4356}(4091,\cdot)\) \(\chi_{4356}(4223,\cdot)\)
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
\((2179,1937,1333)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4356 }(3431, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)