from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,15,8]))
pari: [g,chi] = znchar(Mod(111,1792))
Basic properties
| Modulus: | \(1792\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(307,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bd
\(\chi_{1792}(111,\cdot)\) \(\chi_{1792}(335,\cdot)\) \(\chi_{1792}(559,\cdot)\) \(\chi_{1792}(783,\cdot)\) \(\chi_{1792}(1007,\cdot)\) \(\chi_{1792}(1231,\cdot)\) \(\chi_{1792}(1455,\cdot)\) \(\chi_{1792}(1679,\cdot)\)
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: | 16.16.3484608386920116940487669055488.4 |
Values on generators
\((1023,1541,1025)\) → \((-1,e\left(\frac{15}{16}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1792 }(111, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)