from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2583, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,19]))
pari: [g,chi] = znchar(Mod(8,2583))
Basic properties
| Modulus: | \(2583\) | |
| Conductor: | \(123\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{123}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2583.cy
\(\chi_{2583}(8,\cdot)\) \(\chi_{2583}(197,\cdot)\) \(\chi_{2583}(323,\cdot)\) \(\chi_{2583}(449,\cdot)\) \(\chi_{2583}(512,\cdot)\) \(\chi_{2583}(2276,\cdot)\) \(\chi_{2583}(2339,\cdot)\) \(\chi_{2583}(2465,\cdot)\)
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: | 20.0.259481156482773157712790472689766689.1 |
Values on generators
\((2297,2215,1072)\) → \((-1,1,e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2583 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)