from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,48,5]))
pari: [g,chi] = znchar(Mod(3,8030))
Basic properties
| Modulus: | \(8030\) | |
| Conductor: | \(4015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4015}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8030.es
\(\chi_{8030}(3,\cdot)\) \(\chi_{8030}(487,\cdot)\) \(\chi_{8030}(973,\cdot)\) \(\chi_{8030}(1457,\cdot)\) \(\chi_{8030}(1703,\cdot)\) \(\chi_{8030}(2187,\cdot)\) \(\chi_{8030}(2193,\cdot)\) \(\chi_{8030}(2677,\cdot)\) \(\chi_{8030}(4383,\cdot)\) \(\chi_{8030}(4623,\cdot)\) \(\chi_{8030}(4867,\cdot)\) \(\chi_{8030}(5107,\cdot)\) \(\chi_{8030}(5113,\cdot)\) \(\chi_{8030}(5597,\cdot)\) \(\chi_{8030}(6813,\cdot)\) \(\chi_{8030}(7297,\cdot)\)
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
\((1607,2191,881)\) → \((-i,e\left(\frac{4}{5}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8030 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) |
sage: chi.jacobi_sum(n)