from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([1,8]))
pari: [g,chi] = znchar(Mod(1057,8993))
Basic properties
Modulus: | \(8993\) | |
Conductor: | \(391\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{391}(275,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8993.k
\(\chi_{8993}(1057,\cdot)\) \(\chi_{8993}(1586,\cdot)\) \(\chi_{8993}(2115,\cdot)\) \(\chi_{8993}(3173,\cdot)\) \(\chi_{8993}(6347,\cdot)\) \(\chi_{8993}(7405,\cdot)\) \(\chi_{8993}(7934,\cdot)\) \(\chi_{8993}(8463,\cdot)\)
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: | 16.16.224159169454780289392644342833.1 |
Values on generators
\((530,7940)\) → \((e\left(\frac{1}{16}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8993 }(1057, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)