from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8005, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,11]))
pari: [g,chi] = znchar(Mod(206,8005))
Basic properties
| Modulus: | \(8005\) | |
| Conductor: | \(1601\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1601}(206,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8005.bg
\(\chi_{8005}(206,\cdot)\) \(\chi_{8005}(356,\cdot)\) \(\chi_{8005}(796,\cdot)\) \(\chi_{8005}(1366,\cdot)\) \(\chi_{8005}(1421,\cdot)\) \(\chi_{8005}(1781,\cdot)\) \(\chi_{8005}(1836,\cdot)\) \(\chi_{8005}(2406,\cdot)\) \(\chi_{8005}(2846,\cdot)\) \(\chi_{8005}(2996,\cdot)\) \(\chi_{8005}(3311,\cdot)\) \(\chi_{8005}(3371,\cdot)\) \(\chi_{8005}(5246,\cdot)\) \(\chi_{8005}(5961,\cdot)\) \(\chi_{8005}(7836,\cdot)\) \(\chi_{8005}(7896,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((1602,4806)\) → \((1,e\left(\frac{11}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 8005 }(206, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(-i\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) |
sage: chi.jacobi_sum(n)