from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,10,9,30]))
pari: [g,chi] = znchar(Mod(83,6300))
Basic properties
| Modulus: | \(6300\) | |
| Conductor: | \(6300\) | 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 6300.jb
\(\chi_{6300}(83,\cdot)\) \(\chi_{6300}(167,\cdot)\) \(\chi_{6300}(587,\cdot)\) \(\chi_{6300}(923,\cdot)\) \(\chi_{6300}(1427,\cdot)\) \(\chi_{6300}(1847,\cdot)\) \(\chi_{6300}(2183,\cdot)\) \(\chi_{6300}(2603,\cdot)\) \(\chi_{6300}(2687,\cdot)\) \(\chi_{6300}(3863,\cdot)\) \(\chi_{6300}(3947,\cdot)\) \(\chi_{6300}(4367,\cdot)\) \(\chi_{6300}(4703,\cdot)\) \(\chi_{6300}(5123,\cdot)\) \(\chi_{6300}(5627,\cdot)\) \(\chi_{6300}(5963,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)