from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,15,10,3]))
pari: [g,chi] = znchar(Mod(79,6930))
Basic properties
| Modulus: | \(6930\) | |
| Conductor: | \(3465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3465}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6930.hi
\(\chi_{6930}(79,\cdot)\) \(\chi_{6930}(1339,\cdot)\) \(\chi_{6930}(1579,\cdot)\) \(\chi_{6930}(3229,\cdot)\) \(\chi_{6930}(4099,\cdot)\) \(\chi_{6930}(5359,\cdot)\) \(\chi_{6930}(5749,\cdot)\) \(\chi_{6930}(6619,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((1541,1387,2971,2521)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 6930 }(79, a) \) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(-1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) |
sage: chi.jacobi_sum(n)