from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3264, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,4,8,7]))
pari: [g,chi] = znchar(Mod(113,3264))
Basic properties
| Modulus: | \(3264\) | |
| Conductor: | \(816\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{816}(725,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3264.dx
\(\chi_{3264}(113,\cdot)\) \(\chi_{3264}(209,\cdot)\) \(\chi_{3264}(785,\cdot)\) \(\chi_{3264}(1265,\cdot)\) \(\chi_{3264}(2033,\cdot)\) \(\chi_{3264}(2513,\cdot)\) \(\chi_{3264}(3089,\cdot)\) \(\chi_{3264}(3185,\cdot)\)
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: | 16.16.330387545600365800521582857754247168.2 |
Values on generators
\((511,2245,2177,2689)\) → \((1,i,-1,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3264 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)