from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(53312, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
pari: [g,chi] = znchar(Mod(40769,53312))
Basic properties
| Modulus: | \(53312\) | |
| Conductor: | \(17\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{17}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 53312.hj
\(\chi_{53312}(15681,\cdot)\) \(\chi_{53312}(21953,\cdot)\) \(\chi_{53312}(25089,\cdot)\) \(\chi_{53312}(28225,\cdot)\) \(\chi_{53312}(37633,\cdot)\) \(\chi_{53312}(40769,\cdot)\) \(\chi_{53312}(43905,\cdot)\) \(\chi_{53312}(50177,\cdot)\)
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
\((51647,3333,10881,40769)\) → \((1,1,1,e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 53312 }(40769, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)