from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,2,1]))
pari: [g,chi] = znchar(Mod(37,4005))
Basic properties
| Modulus: | \(4005\) | |
| Conductor: | \(445\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{445}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4005.bf
\(\chi_{4005}(37,\cdot)\) \(\chi_{4005}(433,\cdot)\) \(\chi_{4005}(1387,\cdot)\) \(\chi_{4005}(2593,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.8.691114607742640625.2 |
Values on generators
\((3116,802,181)\) → \((1,i,e\left(\frac{1}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(37, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)