from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1870, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([12,0,9]))
pari: [g,chi] = znchar(Mod(133,1870))
Basic properties
| Modulus: | \(1870\) | |
| Conductor: | \(85\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{85}(48,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1870.bl
\(\chi_{1870}(133,\cdot)\) \(\chi_{1870}(177,\cdot)\) \(\chi_{1870}(243,\cdot)\) \(\chi_{1870}(397,\cdot)\) \(\chi_{1870}(573,\cdot)\) \(\chi_{1870}(683,\cdot)\) \(\chi_{1870}(1167,\cdot)\) \(\chi_{1870}(1387,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.698833752810013621337890625.2 |
Values on generators
\((1497,1531,1431)\) → \((-i,1,e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1870 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)