from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(935, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([12,0,9]))
pari: [g,chi] = znchar(Mod(133,935))
Basic properties
| Modulus: | \(935\) | |
| 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 935.bl
\(\chi_{935}(133,\cdot)\) \(\chi_{935}(177,\cdot)\) \(\chi_{935}(232,\cdot)\) \(\chi_{935}(243,\cdot)\) \(\chi_{935}(397,\cdot)\) \(\chi_{935}(452,\cdot)\) \(\chi_{935}(573,\cdot)\) \(\chi_{935}(683,\cdot)\)
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
\((562,596,496)\) → \((-i,1,e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 935 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-1\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)