from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,22,16]))
pari: [g,chi] = znchar(Mod(2883,8470))
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4235}(2883,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.cc
\(\chi_{8470}(573,\cdot)\) \(\chi_{8470}(1343,\cdot)\) \(\chi_{8470}(1497,\cdot)\) \(\chi_{8470}(2113,\cdot)\) \(\chi_{8470}(2267,\cdot)\) \(\chi_{8470}(2883,\cdot)\) \(\chi_{8470}(3037,\cdot)\) \(\chi_{8470}(3653,\cdot)\) \(\chi_{8470}(3807,\cdot)\) \(\chi_{8470}(4423,\cdot)\) \(\chi_{8470}(4577,\cdot)\) \(\chi_{8470}(5193,\cdot)\) \(\chi_{8470}(5347,\cdot)\) \(\chi_{8470}(5963,\cdot)\) \(\chi_{8470}(6117,\cdot)\) \(\chi_{8470}(6733,\cdot)\) \(\chi_{8470}(6887,\cdot)\) \(\chi_{8470}(7657,\cdot)\) \(\chi_{8470}(8273,\cdot)\) \(\chi_{8470}(8427,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((6777,6051,7141)\) → \((-i,-1,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 8470 }(2883, a) \) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(i\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) |
sage: chi.jacobi_sum(n)