from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3264, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,4]))
pari: [g,chi] = znchar(Mod(13,3264))
Basic properties
Modulus: | \(3264\) | |
Conductor: | \(1088\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1088}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3264.fg
\(\chi_{3264}(13,\cdot)\) \(\chi_{3264}(565,\cdot)\) \(\chi_{3264}(829,\cdot)\) \(\chi_{3264}(1381,\cdot)\) \(\chi_{3264}(1645,\cdot)\) \(\chi_{3264}(2197,\cdot)\) \(\chi_{3264}(2461,\cdot)\) \(\chi_{3264}(3013,\cdot)\)
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: | 16.16.352173532834348866466626446322688851968.1 |
Values on generators
\((511,2245,2177,2689)\) → \((1,e\left(\frac{15}{16}\right),1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3264 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)