from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1020, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,12,7]))
pari: [g,chi] = znchar(Mod(113,1020))
Basic properties
| Modulus: | \(1020\) | |
| Conductor: | \(255\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{255}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1020.ct
\(\chi_{1020}(113,\cdot)\) \(\chi_{1020}(377,\cdot)\) \(\chi_{1020}(437,\cdot)\) \(\chi_{1020}(533,\cdot)\) \(\chi_{1020}(617,\cdot)\) \(\chi_{1020}(653,\cdot)\) \(\chi_{1020}(677,\cdot)\) \(\chi_{1020}(1013,\cdot)\)
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: | Number field defined by a degree 16 polynomial |
Values on generators
\((511,341,817,241)\) → \((1,-1,-i,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1020 }(113, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)