from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,54,5]))
pari: [g,chi] = znchar(Mod(457,4719))
Basic properties
| Modulus: | \(4719\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4719.cg
\(\chi_{4719}(457,\cdot)\) \(\chi_{4719}(475,\cdot)\) \(\chi_{4719}(838,\cdot)\) \(\chi_{4719}(1129,\cdot)\) \(\chi_{4719}(1207,\cdot)\) \(\chi_{4719}(1909,\cdot)\) \(\chi_{4719}(2290,\cdot)\) \(\chi_{4719}(2581,\cdot)\) \(\chi_{4719}(2659,\cdot)\) \(\chi_{4719}(2944,\cdot)\) \(\chi_{4719}(3022,\cdot)\) \(\chi_{4719}(3361,\cdot)\) \(\chi_{4719}(3724,\cdot)\) \(\chi_{4719}(3742,\cdot)\) \(\chi_{4719}(4396,\cdot)\) \(\chi_{4719}(4474,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1574,3511,4357)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4719 }(457, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)