from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,20]))
pari: [g,chi] = znchar(Mod(309,8470))
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 \(\chi_{605}(309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.bo
\(\chi_{8470}(309,\cdot)\) \(\chi_{8470}(1079,\cdot)\) \(\chi_{8470}(1849,\cdot)\) \(\chi_{8470}(2619,\cdot)\) \(\chi_{8470}(4159,\cdot)\) \(\chi_{8470}(4929,\cdot)\) \(\chi_{8470}(5699,\cdot)\) \(\chi_{8470}(6469,\cdot)\) \(\chi_{8470}(7239,\cdot)\) \(\chi_{8470}(8009,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((6777,6051,7141)\) → \((-1,1,e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 8470 }(309, a) \) | \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(-1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)