from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,15]))
pari: [g,chi] = znchar(Mod(8359,8470))
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(494,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.bn
\(\chi_{8470}(659,\cdot)\) \(\chi_{8470}(1429,\cdot)\) \(\chi_{8470}(2199,\cdot)\) \(\chi_{8470}(2969,\cdot)\) \(\chi_{8470}(3739,\cdot)\) \(\chi_{8470}(4509,\cdot)\) \(\chi_{8470}(5279,\cdot)\) \(\chi_{8470}(6819,\cdot)\) \(\chi_{8470}(7589,\cdot)\) \(\chi_{8470}(8359,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.0.243091704711882553644913533390559631408320849609375.1 |
Values on generators
\((6777,6051,7141)\) → \((-1,1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 8470 }(8359, a) \) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(-1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)