from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([7,8]))
pari: [g,chi] = znchar(Mod(82,145))
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.p
\(\chi_{145}(7,\cdot)\) \(\chi_{145}(23,\cdot)\) \(\chi_{145}(52,\cdot)\) \(\chi_{145}(53,\cdot)\) \(\chi_{145}(78,\cdot)\) \(\chi_{145}(82,\cdot)\) \(\chi_{145}(83,\cdot)\) \(\chi_{145}(103,\cdot)\) \(\chi_{145}(107,\cdot)\) \(\chi_{145}(112,\cdot)\) \(\chi_{145}(123,\cdot)\) \(\chi_{145}(132,\cdot)\)
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: | 28.0.59692812354437378574125162510614243030548095703125.1 |
Values on generators
\((117,31)\) → \((i,e\left(\frac{2}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 145 }(82, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(i\) | \(e\left(\frac{25}{28}\right)\) |
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)