from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,36,59]))
pari: [g,chi] = znchar(Mod(31,6039))
Basic properties
| Modulus: | \(6039\) | |
| Conductor: | \(6039\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6039.qc
\(\chi_{6039}(31,\cdot)\) \(\chi_{6039}(148,\cdot)\) \(\chi_{6039}(421,\cdot)\) \(\chi_{6039}(823,\cdot)\) \(\chi_{6039}(1291,\cdot)\) \(\chi_{6039}(2194,\cdot)\) \(\chi_{6039}(2335,\cdot)\) \(\chi_{6039}(2995,\cdot)\) \(\chi_{6039}(3337,\cdot)\) \(\chi_{6039}(3958,\cdot)\) \(\chi_{6039}(4948,\cdot)\) \(\chi_{6039}(5020,\cdot)\) \(\chi_{6039}(5053,\cdot)\) \(\chi_{6039}(5065,\cdot)\) \(\chi_{6039}(5098,\cdot)\) \(\chi_{6039}(5107,\cdot)\)
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
\((1343,5491,5248)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{59}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6039 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(-i\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)