from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,6,47]))
pari: [g,chi] = znchar(Mod(871,6039))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6039.pb
\(\chi_{6039}(871,\cdot)\) \(\chi_{6039}(1030,\cdot)\) \(\chi_{6039}(1438,\cdot)\) \(\chi_{6039}(1942,\cdot)\) \(\chi_{6039}(2020,\cdot)\) \(\chi_{6039}(2527,\cdot)\) \(\chi_{6039}(2560,\cdot)\) \(\chi_{6039}(2983,\cdot)\) \(\chi_{6039}(3643,\cdot)\) \(\chi_{6039}(4219,\cdot)\) \(\chi_{6039}(4252,\cdot)\) \(\chi_{6039}(5155,\cdot)\) \(\chi_{6039}(5431,\cdot)\) \(\chi_{6039}(5557,\cdot)\) \(\chi_{6039}(5935,\cdot)\) \(\chi_{6039}(5947,\cdot)\)
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
\((1343,5491,5248)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right),e\left(\frac{47}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6039 }(871, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(-i\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{43}{60}\right)\) |
sage: chi.jacobi_sum(n)