from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,7,10]))
pari: [g,chi] = znchar(Mod(503,6300))
Basic properties
| Modulus: | \(6300\) | |
| Conductor: | \(2100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2100}(503,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6300.fm
\(\chi_{6300}(503,\cdot)\) \(\chi_{6300}(1763,\cdot)\) \(\chi_{6300}(2267,\cdot)\) \(\chi_{6300}(3023,\cdot)\) \(\chi_{6300}(3527,\cdot)\) \(\chi_{6300}(4283,\cdot)\) \(\chi_{6300}(4787,\cdot)\) \(\chi_{6300}(6047,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.50902957086795043945312500000000000000000000.1 |
Values on generators
\((3151,2801,3277,3601)\) → \((-1,-1,e\left(\frac{7}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(503, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)