from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41650, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
pari: [g,chi] = znchar(Mod(2451,41650))
Basic properties
| Modulus: | \(41650\) | |
| 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: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 41650.cp
\(\chi_{41650}(2451,\cdot)\) \(\chi_{41650}(4901,\cdot)\) \(\chi_{41650}(7351,\cdot)\) \(\chi_{41650}(12251,\cdot)\) \(\chi_{41650}(26951,\cdot)\) \(\chi_{41650}(31851,\cdot)\) \(\chi_{41650}(34301,\cdot)\) \(\chi_{41650}(36751,\cdot)\)
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
\((18327,2551,2451)\) → \((1,1,e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 41650 }(2451, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)