from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(291, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,27]))
pari: [g,chi] = znchar(Mod(19,291))
Basic properties
| Modulus: | \(291\) | |
| Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{97}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 291.t
\(\chi_{291}(19,\cdot)\) \(\chi_{291}(28,\cdot)\) \(\chi_{291}(34,\cdot)\) \(\chi_{291}(46,\cdot)\) \(\chi_{291}(52,\cdot)\) \(\chi_{291}(55,\cdot)\) \(\chi_{291}(67,\cdot)\) \(\chi_{291}(127,\cdot)\) \(\chi_{291}(139,\cdot)\) \(\chi_{291}(142,\cdot)\) \(\chi_{291}(148,\cdot)\) \(\chi_{291}(160,\cdot)\) \(\chi_{291}(166,\cdot)\) \(\chi_{291}(175,\cdot)\) \(\chi_{291}(214,\cdot)\) \(\chi_{291}(271,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((98,199)\) → \((1,e\left(\frac{27}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 291 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(-i\) |
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)