from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,9,40]))
pari: [g,chi] = znchar(Mod(2083,6300))
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.je
\(\chi_{6300}(67,\cdot)\) \(\chi_{6300}(583,\cdot)\) \(\chi_{6300}(823,\cdot)\) \(\chi_{6300}(1087,\cdot)\) \(\chi_{6300}(1327,\cdot)\) \(\chi_{6300}(2083,\cdot)\) \(\chi_{6300}(2347,\cdot)\) \(\chi_{6300}(2587,\cdot)\) \(\chi_{6300}(3103,\cdot)\) \(\chi_{6300}(3847,\cdot)\) \(\chi_{6300}(4363,\cdot)\) \(\chi_{6300}(4603,\cdot)\) \(\chi_{6300}(4867,\cdot)\) \(\chi_{6300}(5623,\cdot)\) \(\chi_{6300}(5863,\cdot)\) \(\chi_{6300}(6127,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(2083, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)