from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,11,8]))
pari: [g,chi] = znchar(Mod(209,1792))
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 \(\chi_{448}(349,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bf
\(\chi_{1792}(209,\cdot)\) \(\chi_{1792}(433,\cdot)\) \(\chi_{1792}(657,\cdot)\) \(\chi_{1792}(881,\cdot)\) \(\chi_{1792}(1105,\cdot)\) \(\chi_{1792}(1329,\cdot)\) \(\chi_{1792}(1553,\cdot)\) \(\chi_{1792}(1777,\cdot)\)
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: | 16.0.3484608386920116940487669055488.4 |
Values on generators
\((1023,1541,1025)\) → \((1,e\left(\frac{11}{16}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1792 }(209, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)