from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2020, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,19]))
pari: [g,chi] = znchar(Mod(363,2020))
Basic properties
| Modulus: | \(2020\) | |
| Conductor: | \(2020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2020.bm
\(\chi_{2020}(363,\cdot)\) \(\chi_{2020}(443,\cdot)\) \(\chi_{2020}(567,\cdot)\) \(\chi_{2020}(647,\cdot)\) \(\chi_{2020}(1067,\cdot)\) \(\chi_{2020}(1483,\cdot)\) \(\chi_{2020}(1547,\cdot)\) \(\chi_{2020}(1963,\cdot)\)
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: | 20.0.3865948641419300830044353557394262780832000000000000000.1 |
Values on generators
\((1011,1617,1921)\) → \((-1,-i,e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2020 }(363, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)