from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8005, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,9]))
pari: [g,chi] = znchar(Mod(219,8005))
Basic properties
| Modulus: | \(8005\) | |
| Conductor: | \(8005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8005.bh
\(\chi_{8005}(219,\cdot)\) \(\chi_{8005}(899,\cdot)\) \(\chi_{8005}(1024,\cdot)\) \(\chi_{8005}(1389,\cdot)\) \(\chi_{8005}(2339,\cdot)\) \(\chi_{8005}(2464,\cdot)\) \(\chi_{8005}(3414,\cdot)\) \(\chi_{8005}(3779,\cdot)\) \(\chi_{8005}(3904,\cdot)\) \(\chi_{8005}(4584,\cdot)\) \(\chi_{8005}(5469,\cdot)\) \(\chi_{8005}(5649,\cdot)\) \(\chi_{8005}(5929,\cdot)\) \(\chi_{8005}(6879,\cdot)\) \(\chi_{8005}(7159,\cdot)\) \(\chi_{8005}(7339,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((1602,4806)\) → \((-1,e\left(\frac{9}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 8005 }(219, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)