from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,4]))
pari: [g,chi] = znchar(Mod(6161,8470))
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(121\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{121}(111,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.bc
\(\chi_{8470}(771,\cdot)\) \(\chi_{8470}(1541,\cdot)\) \(\chi_{8470}(2311,\cdot)\) \(\chi_{8470}(3081,\cdot)\) \(\chi_{8470}(3851,\cdot)\) \(\chi_{8470}(4621,\cdot)\) \(\chi_{8470}(5391,\cdot)\) \(\chi_{8470}(6161,\cdot)\) \(\chi_{8470}(6931,\cdot)\) \(\chi_{8470}(7701,\cdot)\)
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: | 11.11.672749994932560009201.1 |
Values on generators
\((6777,6051,7141)\) → \((1,1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 8470 }(6161, a) \) | \(1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)