from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(82, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([9]))
pari: [g,chi] = znchar(Mod(19,82))
Basic properties
Modulus: | \(82\) | |
Conductor: | \(41\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{41}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 82.h
\(\chi_{82}(7,\cdot)\) \(\chi_{82}(11,\cdot)\) \(\chi_{82}(13,\cdot)\) \(\chi_{82}(15,\cdot)\) \(\chi_{82}(17,\cdot)\) \(\chi_{82}(19,\cdot)\) \(\chi_{82}(29,\cdot)\) \(\chi_{82}(35,\cdot)\) \(\chi_{82}(47,\cdot)\) \(\chi_{82}(53,\cdot)\) \(\chi_{82}(63,\cdot)\) \(\chi_{82}(65,\cdot)\) \(\chi_{82}(67,\cdot)\) \(\chi_{82}(69,\cdot)\) \(\chi_{82}(71,\cdot)\) \(\chi_{82}(75,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\(47\) → \(e\left(\frac{9}{40}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 82 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(-i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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)