from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(97, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(67,97))
Basic properties
Modulus: | \(97\) | |
Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 97.j
\(\chi_{97}(19,\cdot)\) \(\chi_{97}(20,\cdot)\) \(\chi_{97}(28,\cdot)\) \(\chi_{97}(30,\cdot)\) \(\chi_{97}(34,\cdot)\) \(\chi_{97}(42,\cdot)\) \(\chi_{97}(45,\cdot)\) \(\chi_{97}(46,\cdot)\) \(\chi_{97}(51,\cdot)\) \(\chi_{97}(52,\cdot)\) \(\chi_{97}(55,\cdot)\) \(\chi_{97}(63,\cdot)\) \(\chi_{97}(67,\cdot)\) \(\chi_{97}(69,\cdot)\) \(\chi_{97}(77,\cdot)\) \(\chi_{97}(78,\cdot)\)
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
\(5\) → \(e\left(\frac{19}{32}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 97 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(-i\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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)