from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2583, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,4,9]))
pari: [g,chi] = znchar(Mod(296,2583))
Basic properties
| Modulus: | \(2583\) | |
| Conductor: | \(861\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{861}(296,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2583.ch
\(\chi_{2583}(296,\cdot)\) \(\chi_{2583}(1754,\cdot)\) \(\chi_{2583}(2123,\cdot)\) \(\chi_{2583}(2510,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.1375835651323879518968769.1 |
Values on generators
\((2297,2215,1072)\) → \((-1,e\left(\frac{1}{3}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2583 }(296, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)