from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,9,12]))
pari: [g,chi] = znchar(Mod(91,2624))
Basic properties
Modulus: | \(2624\) | |
Conductor: | \(2624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2624.cc
\(\chi_{2624}(91,\cdot)\) \(\chi_{2624}(483,\cdot)\) \(\chi_{2624}(747,\cdot)\) \(\chi_{2624}(1139,\cdot)\) \(\chi_{2624}(1403,\cdot)\) \(\chi_{2624}(1795,\cdot)\) \(\chi_{2624}(2059,\cdot)\) \(\chi_{2624}(2451,\cdot)\)
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: | 16.0.13638793002368463464842100756923359457968128.1 |
Values on generators
\((575,1477,129)\) → \((-1,e\left(\frac{9}{16}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2624 }(91, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)