from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,32]))
pari: [g,chi] = znchar(Mod(451,592))
Basic properties
Modulus: | \(592\) | |
Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 592.bz
\(\chi_{592}(83,\cdot)\) \(\chi_{592}(107,\cdot)\) \(\chi_{592}(123,\cdot)\) \(\chi_{592}(155,\cdot)\) \(\chi_{592}(219,\cdot)\) \(\chi_{592}(275,\cdot)\) \(\chi_{592}(379,\cdot)\) \(\chi_{592}(403,\cdot)\) \(\chi_{592}(419,\cdot)\) \(\chi_{592}(451,\cdot)\) \(\chi_{592}(515,\cdot)\) \(\chi_{592}(571,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((223,149,113)\) → \((-1,-i,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 592 }(451, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
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)