from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,50,18]))
pari: [g,chi] = znchar(Mod(19,8624))
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1232}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.fg
\(\chi_{8624}(19,\cdot)\) \(\chi_{8624}(227,\cdot)\) \(\chi_{8624}(1195,\cdot)\) \(\chi_{8624}(1403,\cdot)\) \(\chi_{8624}(1795,\cdot)\) \(\chi_{8624}(2371,\cdot)\) \(\chi_{8624}(2763,\cdot)\) \(\chi_{8624}(3363,\cdot)\) \(\chi_{8624}(4331,\cdot)\) \(\chi_{8624}(4539,\cdot)\) \(\chi_{8624}(5507,\cdot)\) \(\chi_{8624}(5715,\cdot)\) \(\chi_{8624}(6107,\cdot)\) \(\chi_{8624}(6683,\cdot)\) \(\chi_{8624}(7075,\cdot)\) \(\chi_{8624}(7675,\cdot)\)
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
\((5391,6469,7745,3137)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8624 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)