from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(896, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,9,16]))
pari: [g,chi] = znchar(Mod(9,896))
Basic properties
| Modulus: | \(896\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 896.br
\(\chi_{896}(9,\cdot)\) \(\chi_{896}(25,\cdot)\) \(\chi_{896}(121,\cdot)\) \(\chi_{896}(137,\cdot)\) \(\chi_{896}(233,\cdot)\) \(\chi_{896}(249,\cdot)\) \(\chi_{896}(345,\cdot)\) \(\chi_{896}(361,\cdot)\) \(\chi_{896}(457,\cdot)\) \(\chi_{896}(473,\cdot)\) \(\chi_{896}(569,\cdot)\) \(\chi_{896}(585,\cdot)\) \(\chi_{896}(681,\cdot)\) \(\chi_{896}(697,\cdot)\) \(\chi_{896}(793,\cdot)\) \(\chi_{896}(809,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,645,129)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 896 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)