from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2128, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,12,14]))
pari: [g,chi] = znchar(Mod(1059,2128))
Basic properties
| Modulus: | \(2128\) | |
| Conductor: | \(2128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2128.gt
\(\chi_{2128}(67,\cdot)\) \(\chi_{2128}(611,\cdot)\) \(\chi_{2128}(667,\cdot)\) \(\chi_{2128}(851,\cdot)\) \(\chi_{2128}(963,\cdot)\) \(\chi_{2128}(1059,\cdot)\) \(\chi_{2128}(1131,\cdot)\) \(\chi_{2128}(1675,\cdot)\) \(\chi_{2128}(1731,\cdot)\) \(\chi_{2128}(1915,\cdot)\) \(\chi_{2128}(2027,\cdot)\) \(\chi_{2128}(2123,\cdot)\)
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
\((799,533,913,1009)\) → \((-1,-i,e\left(\frac{1}{3}\right),e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2128 }(1059, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(i\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)