from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,45,50,18]))
pari: [g,chi] = znchar(Mod(173,6930))
Basic properties
| Modulus: | \(6930\) | |
| Conductor: | \(3465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3465}(173,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.ih
\(\chi_{6930}(173,\cdot)\) \(\chi_{6930}(437,\cdot)\) \(\chi_{6930}(563,\cdot)\) \(\chi_{6930}(677,\cdot)\) \(\chi_{6930}(1823,\cdot)\) \(\chi_{6930}(2063,\cdot)\) \(\chi_{6930}(2327,\cdot)\) \(\chi_{6930}(3197,\cdot)\) \(\chi_{6930}(3713,\cdot)\) \(\chi_{6930}(4457,\cdot)\) \(\chi_{6930}(4583,\cdot)\) \(\chi_{6930}(4847,\cdot)\) \(\chi_{6930}(5717,\cdot)\) \(\chi_{6930}(5843,\cdot)\) \(\chi_{6930}(6107,\cdot)\) \(\chi_{6930}(6233,\cdot)\)
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
\((1541,1387,2971,2521)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{5}{6}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 6930 }(173, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(-i\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) |
sage: chi.jacobi_sum(n)