from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,20,3,5]))
pari: [g,chi] = znchar(Mod(1879,6300))
Basic properties
| Modulus: | \(6300\) | |
| Conductor: | \(6300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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 6300.gq
\(\chi_{6300}(619,\cdot)\) \(\chi_{6300}(859,\cdot)\) \(\chi_{6300}(1879,\cdot)\) \(\chi_{6300}(2119,\cdot)\) \(\chi_{6300}(3139,\cdot)\) \(\chi_{6300}(3379,\cdot)\) \(\chi_{6300}(4639,\cdot)\) \(\chi_{6300}(5659,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(1879, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)