from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,21,10]))
pari: [g,chi] = znchar(Mod(887,4719))
Basic properties
| Modulus: | \(4719\) | |
| Conductor: | \(429\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{429}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4719.by
\(\chi_{4719}(887,\cdot)\) \(\chi_{4719}(965,\cdot)\) \(\chi_{4719}(1322,\cdot)\) \(\chi_{4719}(1667,\cdot)\) \(\chi_{4719}(3428,\cdot)\) \(\chi_{4719}(3500,\cdot)\) \(\chi_{4719}(3506,\cdot)\) \(\chi_{4719}(4208,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((1574,3511,4357)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4719 }(887, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)