from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5780, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,12,11]))
pari: [g,chi] = znchar(Mod(653,5780))
Basic properties
Modulus: | \(5780\) | |
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 \(\chi_{85}(58,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5780.bi
\(\chi_{5780}(653,\cdot)\) \(\chi_{5780}(2377,\cdot)\) \(\chi_{5780}(3337,\cdot)\) \(\chi_{5780}(3393,\cdot)\) \(\chi_{5780}(4177,\cdot)\) \(\chi_{5780}(4873,\cdot)\) \(\chi_{5780}(4953,\cdot)\) \(\chi_{5780}(5137,\cdot)\)
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: | 16.16.698833752810013621337890625.1 |
Values on generators
\((2891,1157,581)\) → \((1,-i,e\left(\frac{11}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 5780 }(653, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)