from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,48,35]))
pari: [g,chi] = znchar(Mod(245,4719))
Basic properties
| Modulus: | \(4719\) | |
| Conductor: | \(429\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{429}(245,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4719.cj
\(\chi_{4719}(245,\cdot)\) \(\chi_{4719}(323,\cdot)\) \(\chi_{4719}(977,\cdot)\) \(\chi_{4719}(995,\cdot)\) \(\chi_{4719}(1358,\cdot)\) \(\chi_{4719}(1697,\cdot)\) \(\chi_{4719}(1775,\cdot)\) \(\chi_{4719}(2060,\cdot)\) \(\chi_{4719}(2138,\cdot)\) \(\chi_{4719}(2429,\cdot)\) \(\chi_{4719}(2810,\cdot)\) \(\chi_{4719}(3512,\cdot)\) \(\chi_{4719}(3590,\cdot)\) \(\chi_{4719}(3881,\cdot)\) \(\chi_{4719}(4244,\cdot)\) \(\chi_{4719}(4262,\cdot)\)
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
\((1574,3511,4357)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4719 }(245, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)