from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3264, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,4,0,7]))
pari: [g,chi] = znchar(Mod(79,3264))
Basic properties
Modulus: | \(3264\) | |
Conductor: | \(272\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{272}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3264.dy
\(\chi_{3264}(79,\cdot)\) \(\chi_{3264}(175,\cdot)\) \(\chi_{3264}(751,\cdot)\) \(\chi_{3264}(1231,\cdot)\) \(\chi_{3264}(1999,\cdot)\) \(\chi_{3264}(2479,\cdot)\) \(\chi_{3264}(3055,\cdot)\) \(\chi_{3264}(3151,\cdot)\)
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: | 16.16.50356278859985642512053476261888.2 |
Values on generators
\((511,2245,2177,2689)\) → \((-1,i,1,e\left(\frac{7}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3264 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)