from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,25]))
pari: [g,chi] = znchar(Mod(69,145))
Basic properties
Modulus: | \(145\) | |
Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 145.s
\(\chi_{145}(14,\cdot)\) \(\chi_{145}(19,\cdot)\) \(\chi_{145}(39,\cdot)\) \(\chi_{145}(44,\cdot)\) \(\chi_{145}(69,\cdot)\) \(\chi_{145}(79,\cdot)\) \(\chi_{145}(84,\cdot)\) \(\chi_{145}(89,\cdot)\) \(\chi_{145}(114,\cdot)\) \(\chi_{145}(119,\cdot)\) \(\chi_{145}(124,\cdot)\) \(\chi_{145}(134,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.0.18634854406558377293367533932433545897882080078125.1 |
Values on generators
\((117,31)\) → \((-1,e\left(\frac{25}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 145 }(69, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) |
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)