from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,6,4]))
pari: [g,chi] = znchar(Mod(6863,8512))
Basic properties
| Modulus: | \(8512\) | |
| Conductor: | \(2128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2128}(2075,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.io
\(\chi_{8512}(271,\cdot)\) \(\chi_{8512}(367,\cdot)\) \(\chi_{8512}(719,\cdot)\) \(\chi_{8512}(1263,\cdot)\) \(\chi_{8512}(2607,\cdot)\) \(\chi_{8512}(3855,\cdot)\) \(\chi_{8512}(4527,\cdot)\) \(\chi_{8512}(4623,\cdot)\) \(\chi_{8512}(4975,\cdot)\) \(\chi_{8512}(5519,\cdot)\) \(\chi_{8512}(6863,\cdot)\) \(\chi_{8512}(8111,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((5055,6917,7297,3137)\) → \((-1,i,e\left(\frac{1}{6}\right),e\left(\frac{1}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(6863, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(-i\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)